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Theorem List for Metamath Proof Explorer - 29001-29100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj1039 29001 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   =>    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj1040 29002* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph'  <->  [. j  /  i ]. ph )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ch'  <->  [. j  /  i ]. ch )   =>    |-  ( ch'  <->  ( n  e.  D  /\  f  Fn  n  /\  ph'  /\  ps' ) )
 
Theorembnj1047 29003 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( rh 
 <-> 
 A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  ( et'  <->  [. j  /  i ]. et )   =>    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  et' ) )
 
Theorembnj1049 29004 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  ( et  <->  ( ( th  /\ 
 ta  /\  ch  /\  ze )  ->  z  e.  B ) )   =>    |-  ( A. i  e.  n  et  <->  A. i et )
 
Theorembnj1052 29005* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( et 
 <->  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B ) )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  (
 ( th  /\  ta  /\  ch 
 /\  ze )  ->  (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et ) ) )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1053 29006* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( et 
 <->  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B ) )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  (
 ( th  /\  ta  /\  ch 
 /\  ze )  ->  A. i  e.  n  ( rh  ->  et ) )   =>    |-  ( ( th  /\ 
 ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1071 29007 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( n  e.  D  ->  _E  Fr  n )
 
Theorembnj1083 29008 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  K  =  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( f  e.  K  <->  E. n ch )
 
Theorembnj1090 29009* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ( f  e.  K  /\  i  e. 
 dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  ( et'  <->  [. j  /  i ]. et )   &    |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   &    |-  ( ( th  /\ 
 ta  /\  ch  /\  ze )  ->  A. i E. j
 ( ph0  ->  ( f `  i )  C_  B ) )   =>    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  A. i  e.  n  ( rh  ->  et )
 )
 
Theorembnj1093 29010* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  E. j
 ( ( ( th  /\ 
 ta  /\  ch )  /\  ph0 )  ->  (
 f `  i )  C_  B )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   =>    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  A. i E. j
 ( ph0  ->  ( f `  i )  C_  B ) )
 
Theorembnj1097 29011 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( i  =  (/)  /\  ( ( th  /\  ta  /\  ch )  /\  ph0 ) )  ->  ( f `  i
 )  C_  B )
 
Theorembnj1110 29012* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( si 
 <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   &    |-  ( et'  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j
 )  C_  B )
 )   =>    |- 
 E. j ( ( i  =/=  (/)  /\  (
 ( th  /\  ta  /\  ch )  /\  ph0 ) ) 
 ->  ( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B ) )
 
Theorembnj1112 29013* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ps  <->  A. j ( ( j  e.  om  /\  suc  j  e.  n ) 
 ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj1118 29014* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  D  =  ( om  \  { (/)
 } )   &    |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   &    |-  ( et'  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j
 )  C_  B )
 )   =>    |- 
 E. j ( ( i  =/=  (/)  /\  (
 ( th  /\  ta  /\  ch )  /\  ph0 ) ) 
 ->  ( f `  i
 )  C_  B )
 
Theorembnj1121 29015 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th 
 <->  ( R  FrSe  A  /\  X  e.  A )
 )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i
 ) ) )   &    |-  ( et 
 <->  ( ( f  e.  K  /\  i  e. 
 dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  A. i  e.  n  et )   &    |-  K  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B )
 
Theorembnj1123 29016* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  K  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   &    |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( et'  <->  [. j  /  i ]. et )   =>    |-  ( et'  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j
 )  C_  B )
 )
 
Theorembnj1030 29017* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( et 
 <->  ( ( f  e.  K  /\  i  e. 
 dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  ( ph'  <->  [. j  /  i ]. ph )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   &    |-  ( ch'  <->  [. j  /  i ]. ch )   &    |-  ( th'  <->  [. j  /  i ]. th )   &    |-  ( ta'  <->  [. j  /  i ]. ta )   &    |-  ( ze'  <->  [. j  /  i ]. ze )   &    |-  ( et'  <->  [. j  /  i ]. et )   &    |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1124 29018 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th 
 <->  ( R  FrSe  A  /\  X  e.  A )
 )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( th  /\ 
 ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1133 29019* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ta  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. th ) )   &    |-  (
 ( i  e.  n  /\  ta )  ->  th )   =>    |-  ( ch  ->  A. i  e.  n  th )
 
Theorembnj1128 29020* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( th  <->  ( ch  ->  ( f `  i ) 
 C_  A ) )   &    |-  ( ta  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. th ) )   &    |-  ( ph'  <->  [. j  /  i ]. ph )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   &    |-  ( ch'  <->  [. j  /  i ]. ch )   &    |-  ( th'  <->  [. j  /  i ]. th )   =>    |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
 
Theorembnj1127 29021 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
 
Theorembnj1125 29022 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A  /\  Y  e.  trCl ( X ,  A ,  R ) )  ->  trCl ( Y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
 
Theorembnj1145 29023* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( th  <->  ( ( i  =/=  (/)  /\  i  e.  n  /\  ch )  /\  ( j  e.  n  /\  i  =  suc  j ) ) )   =>    |-  trCl
 ( X ,  A ,  R )  C_  A
 
Theorembnj1147 29024 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  trCl ( X ,  A ,  R )  C_  A
 
Theorembnj1137 29025* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  TrFo ( B ,  A ,  R )
 )
 
Theorembnj1148 29026 Property of  pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 pred ( X ,  A ,  R )  e.  _V )
 
Theorembnj1136 29027* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( R 
 FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
 
Theorembnj1152 29028 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( Y  e.  pred ( X ,  A ,  R ) 
 <->  ( Y  e.  A  /\  Y R X ) )
 
Theorembnj1154 29029* Property of  Fr. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/)  /\  B  e.  _V )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
 
Theorembnj1171 29030 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  B  C_  A )   &    |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 z  e.  B  /\  ( w  e.  A  ->  ( w R z 
 ->  -.  w  e.  B ) ) ) )   =>    |-  E. z A. w ( ( ph  /\  ps )  ->  ( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )
 
Theorembnj1172 29031 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )   &    |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 ( ph  /\  ps  /\  z  e.  C )  /\  ( th  ->  ( w R z  ->  -.  w  e.  B ) ) ) )   &    |-  ( ( ph  /\ 
 ps  /\  z  e.  C )  ->  ( th  <->  w  e.  A ) )   =>    |-  E. z A. w ( ( ph  /\  ps )  ->  ( z  e.  B  /\  ( w  e.  A  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
 
Theorembnj1173 29032 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )   &    |-  ( th 
 <->  ( ( R  FrSe  A 
 /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A ) )   &    |-  ( ( ph  /\ 
 ps )  ->  R  FrSe  A )   &    |-  ( ( ph  /\ 
 ps )  ->  X  e.  A )   =>    |-  ( ( ph  /\  ps  /\  z  e.  C ) 
 ->  ( th  <->  w  e.  A ) )
 
Theorembnj1174 29033 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )   &    |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 z  e.  C  /\  ( th  ->  ( w R z  ->  -.  w  e.  C ) ) ) )   &    |-  ( th  ->  ( w R z  ->  w  e.  trCl ( X ,  A ,  R ) ) )   =>    |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 ( ph  /\  ps  /\  z  e.  C )  /\  ( th  ->  ( w R z  ->  -.  w  e.  B ) ) ) )
 
Theorembnj1175 29034 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  C  =  (  trCl ( X ,  A ,  R )  i^i  B )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  ( w  e.  A  /\  w R z ) ) )   &    |-  ( th  <->  ( ( R 
 FrSe  A  /\  X  e.  A  /\  z  e.  trCl ( X ,  A ,  R ) )  /\  ( R  FrSe  A  /\  z  e.  A )  /\  w  e.  A ) )   =>    |-  ( th  ->  ( w R z  ->  w  e.  trCl ( X ,  A ,  R )
 ) )
 
Theorembnj1176 29035* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e.  _V ) )   &    |-  ( ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e.  _V )  ->  E. z  e.  C  A. w  e.  C  -.  w R z )   =>    |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 z  e.  C  /\  ( th  ->  ( w R z  ->  -.  w  e.  C ) ) ) )
 
Theorembnj1177 29036 Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <->  ( X  e.  B  /\  y  e.  B  /\  y R X ) )   &    |-  C  =  ( 
 trCl ( X ,  A ,  R )  i^i  B )   &    |-  ( ( ph  /\ 
 ps )  ->  R  FrSe  A )   &    |-  ( ( ph  /\ 
 ps )  ->  B  C_  A )   &    |-  ( ( ph  /\ 
 ps )  ->  X  e.  A )   =>    |-  ( ( ph  /\  ps )  ->  ( R  Fr  A  /\  C  C_  A  /\  C  =/=  (/)  /\  C  e.  _V ) )
 
Theorembnj1186 29037* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  E. z A. w ( ( ph  /\ 
 ps )  ->  (
 z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  E. z  e.  B  A. w  e.  B  -.  w R z )
 
Theorembnj1190 29038* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( R  FrSe  A  /\  B  C_  A  /\  B  =/= 
 (/) ) )   &    |-  ( ps 
 <->  ( x  e.  B  /\  y  e.  B  /\  y R x ) )   =>    |-  ( ( ph  /\  ps )  ->  E. w  e.  B  A. z  e.  B  -.  z R w )
 
Theorembnj1189 29039* Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( R  FrSe  A  /\  B  C_  A  /\  B  =/= 
 (/) ) )   &    |-  ( ps 
 <->  ( x  e.  B  /\  y  e.  B  /\  y R x ) )   &    |-  ( ch  <->  A. y  e.  B  -.  y R x )   =>    |-  ( ph  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
 
18.26.3  The existence of a minimal element in certain classes
 
Theorembnj69 29040* Existence of a minimal element in certain classes: if  R is well-founded and set-like on 
A, then every non-empty subclass of  A has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  B  C_  A  /\  B  =/= 
 (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
 
Theorembnj1228 29041* Existence of a minimal element in certain classes: if  R is well-founded and set-like on 
A, then every non-empty subclass of  A has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( w  e.  B  ->  A. x  w  e.  B )   =>    |-  ( ( R  FrSe  A 
 /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
 
18.26.4  Well-founded induction
 
Theorembnj1204 29042* Well-founded induction. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. y  e.  A  ( y R x 
 ->  [. y  /  x ].
 ph ) )   =>    |-  ( ( R 
 FrSe  A  /\  A. x  e.  A  ( ps  ->  ph ) )  ->  A. x  e.  A  ph )
 
Theorembnj1234 29043* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  Y  =  <. x ,  (
 f  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  C  =  { f  |  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  Z  =  <. x ,  (
 g  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  D  =  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) }   =>    |-  C  =  D
 
Theorembnj1245 29044* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   &    |-  Z  =  <. x ,  ( h  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  K  =  { h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  Z ) ) }   =>    |-  ( ph  ->  dom  h  C_  A )
 
Theorembnj1256 29045* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ph  ->  E. d  e.  B  g  Fn  d )
 
Theorembnj1259 29046* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ph  ->  E. d  e.  B  h  Fn  d )
 
Theorembnj1253 29047* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ph  ->  E  =/=  (/) )
 
Theorembnj1279 29048* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x )  ->  (  pred ( x ,  A ,  R )  i^i  E )  =  (/) )
 
Theorembnj1286 29049* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   =>    |-  ( ps  ->  pred
 ( x ,  A ,  R )  C_  D )
 
Theorembnj1280 29050* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   &    |-  ( ps  ->  (  pred ( x ,  A ,  R )  i^i  E )  =  (/) )   =>    |-  ( ps  ->  (
 g  |`  pred ( x ,  A ,  R )
 )  =  ( h  |`  pred ( x ,  A ,  R )
 ) )
 
Theorembnj1296 29051* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   &    |-  E  =  { x  e.  D  |  ( g `  x )  =/=  ( h `  x ) }   &    |-  ( ph 
 <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )   &    |-  ( ps  <->  ( ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )   &    |-  ( ps  ->  ( g  |`  pred
 ( x ,  A ,  R ) )  =  ( h  |`  pred ( x ,  A ,  R ) ) )   &    |-  Z  =  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.   &    |-  K  =  { g  |  E. d  e.  B  (
 g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) }   &    |-  W  =  <. x ,  ( h  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  L  =  { h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) ) }   =>    |-  ( ps  ->  ( g `  x )  =  ( h `  x ) )
 
Theorembnj1309 29052* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   =>    |-  ( w  e.  B  ->  A. x  w  e.  B )
 
Theorembnj1307 29053* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( w  e.  B  ->  A. x  w  e.  B )   =>    |-  ( w  e.  C  ->  A. x  w  e.  C )
 
Theorembnj1311 29054* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   =>    |-  ( ( R 
 FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  (
 g  |`  D )  =  ( h  |`  D ) )
 
Theorembnj1318 29055 Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( X  =  Y  ->  trCl
 ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )
 
Theorembnj1326 29056* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  D  =  ( dom  g  i^i 
 dom  h )   =>    |-  ( ( R 
 FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  (
 g  |`  D )  =  ( h  |`  D ) )
 
Theorembnj1321 29057* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  E. f ta )  ->  E! f ta )
 
Theorembnj1364 29058 Property of  FrSe. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( R  FrSe  A  ->  R  Se  A )
 
Theorembnj1371 29059* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  ( ta'  <->  ( f  e.  C  /\  dom  f  =  ( {
 y }  u.  trCl ( y ,  A ,  R ) ) ) )   =>    |-  ( f  e.  H  ->  Fun  f )
 
Theorembnj1373 29060* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  ( ta'  <->  [. y  /  x ].
 ta )   =>    |-  ( ta'  <->  ( f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
 
Theorembnj1374 29061* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   =>    |-  ( f  e.  H  ->  f  e.  C )
 
Theorembnj1384 29062* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   =>    |-  ( R  FrSe  A  ->  Fun  P )
 
Theorembnj1388 29063* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   =>    |-  ( ch  ->  A. y  e.  pred  ( x ,  A ,  R ) E. f ta' )
 
Theorembnj1398 29064* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  ( th 
 <->  ( ch  /\  z  e.  U_ y  e.  pred  ( x ,  A ,  R ) ( {
 y }  u.  trCl ( y ,  A ,  R ) ) ) )   &    |-  ( et  <->  ( th  /\  y  e.  pred ( x ,  A ,  R )  /\  z  e.  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   =>    |-  ( ch  ->  U_ y  e.  pred  ( x ,  A ,  R )
 ( { y }  u.  trCl ( y ,  A ,  R ) )  =  dom  P )
 
Theorembnj1413 29065* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  B  e.  _V )
 
Theorembnj1408 29066* Technical lemma for bnj1414 29067. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   &    |-  C  =  ( 
 pred ( X ,  A ,  R )  u.  U_ y  e.  trCl  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( R 
 FrSe  A  /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
 
Theorembnj1414 29067* Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  (  pred ( X ,  A ,  R )  u.  U_ y  e.  pred  ( X ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  trCl ( X ,  A ,  R )  =  B )
 
Theorembnj1415 29068* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   =>    |-  ( ch  ->  dom  P  =  trCl ( x ,  A ,  R ) )
 
Theorembnj1416 29069 Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  dom 
 P  =  trCl ( x ,  A ,  R ) )   =>    |-  ( ch  ->  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
 
Theorembnj1418 29070 Property of  pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 y  e.  pred ( x ,  A ,  R )  ->  y R x )
 
Theorembnj1417 29071* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( ph 
 <->  R  FrSe  A )   &    |-  ( ps 
 <->  -.  x  e.  trCl ( x ,  A ,  R ) )   &    |-  ( ch 
 <-> 
 A. y  e.  A  ( y R x 
 ->  [. y  /  x ].
 ps ) )   &    |-  ( th 
 <->  ( ph  /\  x  e.  A  /\  ch )
 )   &    |-  B  =  (  pred ( x ,  A ,  R )  u.  U_ y  e.  pred  ( x ,  A ,  R )  trCl ( y ,  A ,  R ) )   =>    |-  ( ph  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
 
Theorembnj1421 29072* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  Fun 
 P )   &    |-  ( ch  ->  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )   &    |-  ( ch  ->  dom  P  =  trCl ( x ,  A ,  R ) )   =>    |-  ( ch  ->  Fun 
 Q )
 
Theorembnj1444 29073* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   =>    |-  ( rh  ->  A. y rh )
 
Theorembnj1445 29074* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   &    |-  ( si  <->  ( rh  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   &    |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  (
 f `  x )  =  ( G `  Y ) ) )   &    |-  X  =  <. z ,  (
 f  |`  pred ( z ,  A ,  R ) ) >.   =>    |-  ( si  ->  A. d si )
 
Theorembnj1446 29075* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. d ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1447 29076* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. y ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1448 29077* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   =>    |-  ( ( Q `
  z )  =  ( G `  W )  ->  A. f ( Q `
  z )  =  ( G `  W ) )
 
Theorembnj1449 29078* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   =>    |-  ( ze  ->  A. f ze )
 
Theorembnj1442 29079* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   =>    |-  ( et  ->  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1450 29080* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( th  <->  ( ch  /\  z  e.  E )
 )   &    |-  ( et  <->  ( th  /\  z  e.  { x } ) )   &    |-  ( ze 
 <->  ( th  /\  z  e.  trCl ( x ,  A ,  R )
 ) )   &    |-  ( rh  <->  ( ze  /\  f  e.  H  /\  z  e.  dom  f ) )   &    |-  ( si  <->  ( rh  /\  y  e.  pred ( x ,  A ,  R )  /\  f  e.  C  /\  dom  f  =  ( { y }  u.  trCl
 ( y ,  A ,  R ) ) ) )   &    |-  ( ph  <->  ( si  /\  d  e.  B  /\  f  Fn  d  /\  A. x  e.  d  (
 f `  x )  =  ( G `  Y ) ) )   &    |-  X  =  <. z ,  (
 f  |`  pred ( z ,  A ,  R ) ) >.   =>    |-  ( ze  ->  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1423 29081* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  P  Fn  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  Fn  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )   =>    |-  ( ch  ->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )
 
Theorembnj1452 29082* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   =>    |-  ( ch  ->  E  e.  B )
 
Theorembnj1466 29083* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( w  e.  Q  ->  A. f  w  e.  Q )
 
Theorembnj1467 29084* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( w  e.  Q  ->  A. d  w  e.  Q )
 
Theorembnj1463 29085* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   &    |-  ( ch  ->  Q  e.  _V )   &    |-  ( ch  ->  A. z  e.  E  ( Q `  z )  =  ( G `  W ) )   &    |-  ( ch  ->  Q  Fn  E )   &    |-  ( ch  ->  E  e.  B )   =>    |-  ( ch  ->  Q  e.  C )
 
Theorembnj1489 29086* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   =>    |-  ( ch  ->  Q  e.  _V )
 
Theorembnj1491 29087* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  ( ch  ->  ( Q  e.  C  /\  dom 
 Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )   =>    |-  ( ( ch  /\  Q  e.  _V )  ->  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
 
Theorembnj1312 29088* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  ( ta 
 <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl
 ( x ,  A ,  R ) ) ) )   &    |-  D  =  { x  e.  A  |  -.  E. f ta }   &    |-  ( ps 
 <->  ( R  FrSe  A  /\  D  =/=  (/) ) )   &    |-  ( ch 
 <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )   &    |-  ( ta'  <->  [. y  /  x ]. ta )   &    |-  H  =  {
 f  |  E. y  e.  pred  ( x ,  A ,  R ) ta'
 }   &    |-  P  =  U. H   &    |-  Z  =  <. x ,  ( P  |`  pred ( x ,  A ,  R )
 ) >.   &    |-  Q  =  ( P  u.  { <. x ,  ( G `  Z )
 >. } )   &    |-  W  =  <. z ,  ( Q  |`  pred (
 z ,  A ,  R ) ) >.   &    |-  E  =  ( { x }  u.  trCl ( x ,  A ,  R )
 )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
 
Theorembnj1493 29089* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R )
 ) )
 
Theorembnj1497 29090* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  A. g  e.  C  Fun  g
 
Theorembnj1498 29091* Technical lemma for bnj60 29092. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  dom 
 F  =  A )
 
18.26.5  Well-founded recursion, part 1 of 3
 
Theorembnj60 29092* Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  F  Fn  A )
 
Theorembnj1514 29093* Technical lemma for bnj1500 29098. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   =>    |-  (
 f  e.  C  ->  A. x  e.  dom  f
 ( f `  x )  =  ( G `  Y ) )
 
Theorembnj1518 29094* Technical lemma for bnj1500 29098. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   =>    |-  ( ps  ->  A. d ps )
 
Theorembnj1519 29095* Technical lemma for bnj1500 29098. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. d
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1520 29096* Technical lemma for bnj1500 29098. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. )  ->  A. f
 ( F `  x )  =  ( G ` 
 <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1501 29097* Technical lemma for bnj1500 29098. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  x  e.  A ) )   &    |-  ( ps  <->  ( ph  /\  f  e.  C  /\  x  e. 
 dom  f ) )   &    |-  ( ch  <->  ( ps  /\  d  e.  B  /\  dom  f  =  d ) )   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
18.26.6  Well-founded recursion, part 2 of 3
 
Theorembnj1500 29098* Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   =>    |-  ( R  FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )
 
Theorembnj1525 29099* Technical lemma for bnj1522 29102. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d ) }   &    |-  Y  =  <. x ,  ( f  |`  pred
 ( x ,  A ,  R ) ) >.   &    |-  C  =  { f  |  E. d  e.  B  (
 f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }   &    |-  F  =  U. C   &    |-  ( ph  <->  ( R  FrSe  A 
 /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R )
 ) >. ) ) )   &    |-  ( ps  <->  ( ph  /\  F  =/=  H ) )   =>    |-  ( ps  ->  A. x ps )
 
Theorembnj1529 29100* Technical lemma for bnj1522 29102. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R )
 ) >. ) )   &    |-  ( w  e.  F  ->  A. x  w  e.  F )   =>    |-  ( ch  ->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
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