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Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj1534 29201* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }   &    |-  ( w  e.  F  ->  A. x  w  e.  F )   =>    |-  D  =  { z  e.  A  |  ( F `
  z )  =/=  ( H `  z
 ) }
 
Theorembnj1536 29202* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  ( ph  ->  B 
 C_  A )   &    |-  ( ph  ->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )   =>    |-  ( ph  ->  ( F  |`  B )  =  ( G  |`  B ) )
 
Theorembnj1538 29203 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  =  { x  e.  B  |  ph }   =>    |-  ( x  e.  A  -> 
 ph )
 
Theorembnj1541 29204 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( ps  /\  A  =/=  B ) )   &    |-  -.  ph   =>    |-  ( ps  ->  A  =  B )
 
Theorembnj1542 29205* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   &    |-  ( ph  ->  F  =/=  G )   &    |-  ( w  e.  F  ->  A. x  w  e.  F )   =>    |-  ( ph  ->  E. x  e.  A  ( F `  x )  =/=  ( G `  x ) )
 
18.26.2  Well founded induction and recursion
 
Theorembnj110 29206* Well-founded induction restricted to a set ( A  e.  _V). 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.)
 |-  A  e.  _V   &    |-  ( ps  <->  A. y  e.  A  ( y R x 
 ->  [. y  /  x ].
 ph ) )   =>    |-  ( ( R  Fr  A  /\  A. x  e.  A  ( ps  ->  ph ) )  ->  A. x  e.  A  ph )
 
Theorembnj157 29207* Well-founded induction restricted to a set ( A  e.  _V). 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 ) )   &    |-  A  e.  _V   &    |-  R  Fr  A   =>    |-  ( A. x  e.  A  ( ps  ->  ph )  ->  A. x  e.  A  ph )
 
Theorembnj66 29208* Technical lemma for bnj60 29408. 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 ) ) }   =>    |-  (
 g  e.  C  ->  Rel  g )
 
Theorembnj91 29209* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  Z  e.  _V   =>    |-  ( [. Z  /  y ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj92 29210* First-order logic and set theory. (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 ) ) )   &    |-  Z  e.  _V   =>    |-  ( [. Z  /  n ].
 ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  Z  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj93 29211* Technical lemma for bnj97 29214. 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.)
 |-  (
 ( R  FrSe  A  /\  x  e.  A )  -> 
 pred ( x ,  A ,  R )  e.  _V )
 
Theorembnj95 29212 Technical lemma for bnj124 29219. 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.)
 |-  F  =  { <. (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  F  e.  _V
 
Theorembnj96 29213* Technical lemma for bnj150 29224. 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.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
 |-  F  =  { <. (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
 
Theorembnj97 29214* Technical lemma for bnj150 29224. 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.)
 |-  F  =  { <. (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj98 29215 Technical lemma for bnj150 29224. 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.)
 |-  A. i  e.  om  ( suc  i  e.  1o  ->  ( F ` 
 suc  i )  = 
 U_ y  e.  ( F `  i )  pred ( y ,  A ,  R ) )
 
Theorembnj106 29216* First-order logic and set theory. (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 ) ) )   &    |-  F  e.  _V   =>    |-  ( [. F  /  f ]. [. 1o  /  n ].
 ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj118 29217* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ph'  <->  [. 1o  /  n ].
 ph )   =>    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj121 29218* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ze 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )   &    |-  ( ze'  <->  [. 1o  /  n ]. ze )   &    |-  ( ph'  <->  [. 1o  /  n ].
 ph )   &    |-  ( ps'  <->  [. 1o  /  n ].
 ps )   =>    |-  ( ze'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  (
 f  Fn  1o  /\  ph'  /\  ps' ) ) )
 
Theorembnj124 29219* Technical lemma for bnj150 29224. 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.)
 |-  F  =  { <. (/) ,  pred ( x ,  A ,  R ) >. }   &    |-  ( ph"  <->  [. F  /  f ]. ph' )   &    |-  ( ps"  <->  [. F  /  f ]. ps' )   &    |-  ( ze"  <->  [. F  /  f ]. ze' )   &    |-  ( ze'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   =>    |-  ( ze"  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  ( F  Fn  1o  /\  ph"  /\  ps" ) ) )
 
Theorembnj125 29220* Technical lemma for bnj150 29224. 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 )
 )   &    |-  ( ph'  <->  [. 1o  /  n ].
 ph )   &    |-  ( ph"  <->  [. F  /  f ]. ph' )   &    |-  F  =  { <.
 (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  ( ph"  <->  ( F `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj126 29221* Technical lemma for bnj150 29224. 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'  <->  [. 1o  /  n ].
 ps )   &    |-  ( ps"  <->  [. F  /  f ]. ps' )   &    |-  F  =  { <.
 (/) ,  pred ( x ,  A ,  R ) >. }   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj130 29222* Technical lemma for bnj151 29225. 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 )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ph'  <->  [. 1o  /  n ].
 ph )   &    |-  ( ps'  <->  [. 1o  /  n ].
 ps )   &    |-  ( th'  <->  [. 1o  /  n ].
 th )   =>    |-  ( th'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn 
 1o  /\  ph'  /\  ps' ) ) )
 
Theorembnj149 29223* Technical lemma for bnj151 29225. 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.)
 |-  ( th1  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   &    |-  ( ze0  <->  ( f  Fn 
 1o  /\  ph'  /\  ps' ) )   &    |-  ( ze1  <->  [. g  /  f ]. ze0 )   &    |-  ( ph1  <->  [. g  /  f ]. ph' )   &    |-  ( ps1  <->  [. g  /  f ]. ps' )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   =>    |-  th1
 
Theorembnj150 29224* Technical lemma for bnj151 29225. 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 ) ) )   &    |-  ( ze  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  (
 f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ph'  <->  [. 1o  /  n ].
 ph )   &    |-  ( ps'  <->  [. 1o  /  n ].
 ps )   &    |-  ( th0  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E. f
 ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   &    |-  ( ze'  <->  [. 1o  /  n ]. ze )   &    |-  F  =  { <.
 (/) ,  pred ( x ,  A ,  R ) >. }   &    |-  ( ph"  <->  [. F  /  f ]. ph' )   &    |-  ( ps"  <->  [. F  /  f ]. ps' )   &    |-  ( ze"  <->  [. F  /  f ]. ze' )   =>    |-  th0
 
Theorembnj151 29225* Technical lemma for bnj153 29228. 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  \  { (/) } )   &    |-  ( th 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ta  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 th ) )   &    |-  ( ze 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )   &    |-  ( ph'  <->  [. 1o  /  n ]. ph )   &    |-  ( ps'  <->  [. 1o  /  n ].
 ps )   &    |-  ( th'  <->  [. 1o  /  n ].
 th )   &    |-  ( th0  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E. f
 ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   &    |-  ( th1  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E* f ( f  Fn  1o  /\  ph'  /\  ps' ) ) )   &    |-  ( ze'  <->  [. 1o  /  n ].
 ze )   &    |-  F  =  { <.
 (/) ,  pred ( x ,  A ,  R ) >. }   &    |-  ( ph"  <->  [. F  /  f ]. ph' )   &    |-  ( ps"  <->  [. F  /  f ]. ps' )   &    |-  ( ze"  <->  [. F  /  f ]. ze' )   &    |-  ( ze0  <->  ( f  Fn 
 1o  /\  ph'  /\  ps' ) )   &    |-  ( ze1  <->  [. g  /  f ]. ze0 )   &    |-  ( ph1  <->  [. g  /  f ]. ph' )   &    |-  ( ps1  <->  [. g  /  f ]. ps' )   =>    |-  ( n  =  1o  ->  ( ( n  e.  D  /\  ta )  ->  th ) )
 
Theorembnj154 29226* Technical lemma for bnj153 29228. 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.)
 |-  ( ph1  <->  [. g  /  f ]. ph' )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   =>    |-  ( ph1  <->  ( g `  (/) )  =  pred ( x ,  A ,  R ) )
 
Theorembnj155 29227* Technical lemma for bnj153 29228. 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.)
 |-  ( ps1  <->  [. g  /  f ]. ps' )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ps1  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj153 29228* Technical lemma for bnj852 29269. 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  \  { (/) } )   &    |-  ( th 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ta  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 th ) )   =>    |-  ( n  =  1o  ->  ( ( n  e.  D  /\  ta )  ->  th )
 )
 
Theorembnj207 29229* Technical lemma for bnj852 29269. 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 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( ph'  <->  [. M  /  n ].
 ph )   &    |-  ( ps'  <->  [. M  /  n ].
 ps )   &    |-  ( ch'  <->  [. M  /  n ].
 ch )   &    |-  M  e.  _V   =>    |-  ( ch'  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  M  /\  ph'  /\  ps' ) ) )
 
Theorembnj213 29230 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  pred ( X ,  A ,  R )  C_  A
 
Theorembnj222 29231* Technical lemma for bnj229 29232. 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. m  e.  om  ( suc  m  e.  N  ->  ( F `  suc  m )  =  U_ y  e.  ( F `  m )  pred ( y ,  A ,  R ) ) )
 
Theorembnj229 29232* Technical lemma for bnj517 29233. 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 ) ) )   =>    |-  ( ( n  e.  N  /\  ( suc 
 m  =  n  /\  m  e.  om  /\  ps ) )  ->  ( F `
  n )  C_  A )
 
Theorembnj517 29233* Technical lemma for bnj518 29234. 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 
 <->  ( 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 ) ) )   =>    |-  ( ( N  e.  om 
 /\  ph  /\  ps )  ->  A. n  e.  N  ( F `  n ) 
 C_  A )
 
Theorembnj518 29234* Technical lemma for bnj852 29269. 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 ) ) )   &    |-  ( ta  <->  ( ph  /\  ps  /\  n  e.  om  /\  p  e.  n )
 )   =>    |-  ( ( R  FrSe  A 
 /\  ta )  ->  A. y  e.  ( f `  p )  pred ( y ,  A ,  R )  e.  _V )
 
Theorembnj523 29235* Technical lemma for bnj852 29269. 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 )
 )   &    |-  ( ph'  <->  [. M  /  n ].
 ph )   &    |-  M  e.  _V   =>    |-  ( ph'  <->  ( F `  (/) )  = 
 pred ( X ,  A ,  R )
 )
 
Theorembnj526 29236* Technical lemma for bnj852 29269. 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 )
 )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  G  e.  _V   =>    |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
 
Theorembnj528 29237 Technical lemma for bnj852 29269. 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.)
 |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   =>    |-  G  e.  _V
 
Theorembnj535 29238* Technical lemma for bnj852 29269. 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.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m )  ->  G  Fn  n )
 
Theorembnj539 29239* Technical lemma for bnj852 29269. 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'  <->  [. M  /  n ].
 ps )   &    |-  M  e.  _V   =>    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  M  ->  ( F `  suc  i
 )  =  U_ y  e.  ( F `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj540 29240* Technical lemma for bnj852 29269. 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"  <->  [. G  /  f ]. ps )   &    |-  G  e.  _V   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj543 29241* Technical lemma for bnj852 29269. 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.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  e.  m ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
 
Theorembnj544 29242* Technical lemma for bnj852 29269. 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.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
 
Theorembnj545 29243 Technical lemma for bnj852 29269. 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 )
 )   &    |-  D  =  ( om  \  { (/) } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  G  Fn  n )   &    |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  ph" )
 
Theorembnj546 29244* Technical lemma for bnj852 29269. 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  \  { (/)
 } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  si )  -> 
 U_ y  e.  (
 f `  p )  pred ( y ,  A ,  R )  e.  _V )
 
Theorembnj548 29245* Technical lemma for bnj852 29269. 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.)
 |-  ( ta 
 <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  B  =  U_ y  e.  (
 f `  i )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  G  Fn  n )   =>    |-  ( ( ( R 
 FrSe  A  /\  ta  /\  si )  /\  i  e.  m )  ->  B  =  K )
 
Theorembnj553 29246* Technical lemma for bnj852 29269. 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.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  C  =  U_ y  e.  (
 f `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  B  =  U_ y  e.  (
 f `  i )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i )  pred (
 y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred (
 y ,  A ,  R )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   =>    |-  ( ( ( R 
 FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i )  ->  ( G `  m )  =  L )
 
Theorembnj554 29247* Technical lemma for bnj852 29269. 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 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   =>    |-  ( ( et  /\  ze )  ->  ( ( G `  m )  =  L  <->  ( G `  suc  i )  =  K ) )
 
Theorembnj556 29248 Technical lemma for bnj852 29269. 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.)
 |-  ( si 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  m ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   =>    |-  ( et  ->  si )
 
Theorembnj557 29249* Technical lemma for bnj852 29269. 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  \  { (/)
 } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et  /\  ze )  ->  ( G `  m )  =  L )
 
Theorembnj558 29250* Technical lemma for bnj852 29269. 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  \  { (/)
 } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et  /\  ze )  ->  ( G ` 
 suc  i )  =  K )
 
Theorembnj561 29251 Technical lemma for bnj852 29269. 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.)
 |-  ( si 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  m ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  G  Fn  n )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et )  ->  G  Fn  n )
 
Theorembnj562 29252 Technical lemma for bnj852 29269. 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.)
 |-  ( si 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  m ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  ph" )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  et )  ->  ph" )
 
Theorembnj570 29253* Technical lemma for bnj852 29269. 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  \  { (/)
 } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  ( rh 
 <->  ( i  e.  om  /\ 
 suc  i  e.  n  /\  m  =/=  suc  i
 ) )   &    |-  K  =  U_ y  e.  ( G `  i )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  G  Fn  n )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et  /\  rh )  ->  ( G `
  suc  i )  =  K )
 
Theorembnj571 29254* Technical lemma for bnj852 29269. 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  \  { (/)
 } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   &    |-  ( ( R  FrSe  A 
 /\  ta  /\  et )  ->  G  Fn  n )   &    |-  (
 ps" 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et )  ->  ps" )
 
Theorembnj605 29255* Technical lemma. 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 
 <-> 
 A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph"  <->  [. f  /  f ]. ph )   &    |-  ( ps"  <->  [. f  /  f ]. ps )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  f  e.  _V   &    |-  ( ch'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) ) )   &    |-  ( 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 ) ) )   &    |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )   &    |-  (
 ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  f  Fn  n )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  et )  ->  ph" )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  ps" )   =>    |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E. f ( f  Fn  n  /\  ph  /\  ps ) ) )
 
Theorembnj581 29256* Technical lemma for bnj580 29261. 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). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   &    |-  ( ch'  <->  [. g  /  f ]. ch )   =>    |-  ( ch'  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )
 
Theorembnj589 29257* Technical lemma for bnj852 29269. 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. k  e.  om  ( suc  k  e.  n  ->  ( f `  suc  k )  =  U_ y  e.  ( f `  k
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj590 29258 Technical lemma for bnj852 29269. 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 ) ) )   =>    |-  ( ( B  =  suc  i  /\  ps )  ->  ( i  e.  om  ->  ( B  e.  n  ->  ( f `  B )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) ) )
 
Theorembnj591 29259* Technical lemma for bnj852 29269. 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 
 <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )   =>    |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )
 
Theorembnj594 29260* Technical lemma for bnj852 29269. 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  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ph'  <->  ( g `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch'  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )   &    |-  ( th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )   &    |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )   &    |-  ( ta  <->  A. k  e.  n  ( k  _E  j  -> 
 [. k  /  j ]. th ) )   =>    |-  ( ( j  e.  n  /\  ta )  ->  th )
 
Theorembnj580 29261* Technical lemma for bnj579 29262. 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  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   &    |-  ( ch'  <->  [. g  /  f ]. ch )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( th 
 <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )   &    |-  ( ta  <->  A. k  e.  n  ( k  _E  j  -> 
 [. k  /  j ]. th ) )   =>    |-  ( n  e.  D  ->  E* f ch )
 
Theorembnj579 29262* Technical lemma for bnj852 29269. 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  \  { (/) } )   =>    |-  ( n  e.  D  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) )
 
Theorembnj602 29263 Equality theorem for the  pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( X  =  Y  ->  pred
 ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
 
Theorembnj607 29264* Technical lemma for bnj852 29269. 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 
 <-> 
 A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  G  e.  _V   &    |-  ( ch'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) ) )   &    |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )   &    |-  (
 ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  G  Fn  n )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  et )  ->  ph" )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  ps" )   &    |-  ( 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 ) ) )   &    |-  ( ph0  <->  [. h  /  f ]. ph )   &    |-  ( ps0  <->  [. h  /  f ]. ps )   &    |-  ( ph1  <->  [. G  /  h ]. ph0 )   &    |-  ( ps1  <->  [. G  /  h ]. ps0 )   =>    |-  ( ( n  =/= 
 1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E. f ( f  Fn  n  /\  ph  /\  ps ) ) )
 
Theorembnj609 29265* Technical lemma for bnj852 29269. 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 )
 )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  G  e.  _V   =>    |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
 
Theorembnj611 29266* Technical lemma for bnj852 29269. 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"  <->  [. G  /  f ]. ps )   &    |-  G  e.  _V   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj600 29267* Technical lemma for bnj852 29269. 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  \  { (/) } )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( th  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph'  <->  [. m  /  n ]. ph )   &    |-  ( ps'  <->  [. m  /  n ].
 ps )   &    |-  ( ch'  <->  [. m  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  ( ch"  <->  [. G  /  f ]. ch )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   =>    |-  ( n  =/=  1o  ->  ( ( n  e.  D  /\  th )  ->  ch ) )
 
Theorembnj601 29268* Technical lemma for bnj852 29269. 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  \  { (/) } )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( th  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   =>    |-  ( n  =/= 
 1o  ->  ( ( n  e.  D  /\  th )  ->  ch ) )
 
Theorembnj852 29269* Technical lemma for bnj69 29356. 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  \  { (/) } )   =>    |-  (
 ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
 
Theorembnj864 29270* Technical lemma for bnj69 29356. 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  \  { (/) } )   &    |-  ( ch 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
 )   &    |-  ( th  <->  ( f  Fn  n  /\  ph  /\  ps ) )   =>    |-  ( ch  ->  E! f th )
 
Theorembnj865 29271* Technical lemma for bnj69 29356. 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  \  { (/) } )   &    |-  ( ch 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
 )   &    |-  ( th  <->  ( f  Fn  n  /\  ph  /\  ps ) )   =>    |- 
 E. w A. n ( ch  ->  E. f  e.  w  th )
 
Theorembnj873 29272* Technical lemma for bnj69 29356. 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  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   =>    |-  B  =  { g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }
 
Theorembnj849 29273* Technical lemma for bnj69 29356. 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 
 <->  ( 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 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
 )   &    |-  ( th  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   &    |-  ( th'  <->  [. g  /  f ]. th )   &    |-  ( ta  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  B  e.  _V )
 
Theorembnj882 29274* Definition (using hypotheses for readability) of the function giving the transitive closure of  X in  A by  R. (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 ) }   =>    |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f
 ( f `  i
 )
 
Theorembnj18eq1 29275 Equality theorem for transitive closure. (Contributed by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( X  =  Y  ->  trCl
 ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )
 
Theorembnj893 29276 Property of  trCl. Under certain conditions, the transitive closure of  X in  A by  R is a set. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 trCl ( X ,  A ,  R )  e.  _V )
 
Theorembnj900 29277* Technical lemma for bnj69 29356. 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  \  { (/)
 } )   &    |-  B  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( f  e.  B  -> 
 (/)  e.  dom  f )
 
Theorembnj906 29278 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 pred ( X ,  A ,  R )  C_ 
 trCl ( X ,  A ,  R )
 )
 
Theorembnj908 29279* Technical lemma for bnj69 29356. 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  \  { (/) } )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( th  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph'  <->  [. m  /  n ]. ph )   &    |-  ( ps'  <->  [. m  /  n ].
 ps )   &    |-  ( ch'  <->  [. m  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  ( ch"  <->  [. G  /  f ]. ch )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   =>    |-  ( ( R  FrSe  A 
 /\  x  e.  A  /\  ch'  /\  et )  ->  E. f ( G  Fn  n  /\  ph"  /\  ps" ) )
 
Theorembnj911 29280* Technical lemma for bnj69 29356. 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 ) ) )   =>    |-  ( ( f  Fn  n  /\  ph  /\  ps )  ->  A. i ( f  Fn  n  /\  ph  /\  ps ) )
 
Theorembnj916 29281* Technical lemma for bnj69 29356. 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 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   =>    |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( n  e.  D  /\  ch 
 /\  i  e.  n  /\  y  e.  (
 f `  i )
 ) )
 
Theorembnj917 29282* Technical lemma for bnj69 29356. 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 )
 )   =>    |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i
 ) ) )
 
Theorembnj934 29283* Technical lemma for bnj69 29356. 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 )
 )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  G  e.  _V   =>    |-  (
 ( ph  /\  ( G `
  (/) )  =  ( f `  (/) ) ) 
 ->  ph" )
 
Theorembnj929 29284* Technical lemma for bnj69 29356. 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 )
 )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  C  e.  _V   =>    |-  (
 ( n  e.  D  /\  p  =  suc  n 
 /\  f  Fn  n  /\  ph )  ->  ph" )
 
Theorembnj938 29285* Technical lemma for bnj69 29356. 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  \  { (/)
 } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A  /\  ta  /\  si )  -> 
 U_ y  e.  (
 f `  p )  pred ( y ,  A ,  R )  e.  _V )
 
Theorembnj944 29286* Technical lemma for bnj69 29356. 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 ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ta 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( si  <->  ( n  e.  D  /\  p  = 
 suc  n  /\  m  e.  n ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  ph" )
 
Theorembnj953 29287 Technical lemma for bnj69 29356. 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 ) ) )   &    |-  ( ( G `
  i )  =  ( f `  i
 )  ->  A. y ( G `  i )  =  ( f `  i ) )   =>    |-  ( ( ( G `  i )  =  ( f `  i )  /\  ( G `
  suc  i )  =  ( f `  suc  i )  /\  ( i  e.  om  /\  suc  i  e.  n )  /\  ps )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) )
 
Theorembnj958 29288* Technical lemma for bnj69 29356. 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  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  (
 ( G `  i
 )  =  ( f `
  i )  ->  A. y ( G `  i )  =  (
 f `  i )
 )
 
Theorembnj1000 29289* Technical lemma for bnj852 29269. 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"  <->  [. G  /  f ]. ps )   &    |-  G  e.  _V   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj965 29290* Technical lemma for bnj852 29269. 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"  <->  [. G  /  f ]. ps )   &    |-  C  =  U_ y  e.  ( f `  m )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj964 29291* Technical lemma for bnj69 29356. 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 ) )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  C  =  U_ y  e.  ( f `  m )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( ( R  FrSe  A 
 /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) 
 /\  ( i  e. 
 om  /\  suc  i  e.  p  /\  suc  i  e.  n ) )  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) )   &    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  ( G ` 
 suc  i )  = 
 U_ y  e.  ( G `  i )  pred ( y ,  A ,  R ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  ps" )
 
Theorembnj966 29292* Technical lemma for bnj69 29356. 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  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( ( R  FrSe  A 
 /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )   &    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  G  Fn  p )   =>    |-  ( ( ( R 
 FrSe  A  /\  X  e.  A )  /\  ( ch 
 /\  n  =  suc  m 
 /\  p  =  suc  n )  /\  ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  ( G ` 
 suc  i )  = 
 U_ y  e.  ( G `  i )  pred ( y ,  A ,  R ) )
 
Theorembnj967 29293* Technical lemma for bnj69 29356. 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 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( ( R  FrSe  A 
 /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )   =>    |-  ( ( ( R 
 FrSe  A  /\  X  e.  A )  /\  ( ch 
 /\  n  =  suc  m 
 /\  p  =  suc  n )  /\  ( i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n ) )  ->  ( G ` 
 suc  i )  = 
 U_ y  e.  ( G `  i )  pred ( y ,  A ,  R ) )
 
Theorembnj969 29294* Technical lemma for bnj69 29356. 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 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  ( ta  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( si  <->  ( n  e.  D  /\  p  = 
 suc  n  /\  m  e.  n ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  C  e.  _V )
 
Theorembnj970 29295 Technical lemma for bnj69 29356. 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  \  { (/) } )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  p  e.  D )
 
Theorembnj910 29296* Technical lemma for bnj69 29356. 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 ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ta 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( si  <->  ( n  e.  D  /\  p  = 
 suc  n  /\  m  e.  n ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  ch" )
 
Theorembnj978 29297* Technical lemma for bnj69 29356. 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  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e.  pred ( y ,  A ,  R ) ) )   &    |-  ( th  ->  z  e.  trCl
 ( X ,  A ,  R ) )   =>    |-  ( ( R 
 FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
 
Theorembnj981 29298* Technical lemma for bnj69 29356. 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 )
 )   =>    |-  ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
 ) ) )
 
Theorembnj983 29299* Technical lemma for bnj69 29356. 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 )
 )   =>    |-  ( Z  e.  trCl ( X ,  A ,  R )  <->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) )
 
Theorembnj984 29300 Technical lemma for bnj69 29356. 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 )
 )   &    |-  B  =  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( G  e.  A  ->  ( G  e.  B  <->  [. G  /  f ]. E. n ch ) )
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