Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1030 Unicode version

Theorem bnj1030 29017
Description: 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.)
Hypotheses
Ref Expression
bnj1030.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj1030.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj1030.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj1030.4  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
bnj1030.5  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
bnj1030.6  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
bnj1030.7  |-  D  =  ( om  \  { (/)
} )
bnj1030.8  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
bnj1030.9  |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  (
f `  i )  C_  B ) )
bnj1030.10  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
bnj1030.11  |-  ( ph'  <->  [. j  /  i ]. ph )
bnj1030.12  |-  ( ps'  <->  [. j  /  i ]. ps )
bnj1030.13  |-  ( ch'  <->  [. j  /  i ]. ch )
bnj1030.14  |-  ( th'  <->  [. j  / 
i ]. th )
bnj1030.15  |-  ( ta'  <->  [. j  /  i ]. ta )
bnj1030.16  |-  ( ze'  <->  [. j  /  i ]. ze )
bnj1030.17  |-  ( et'  <->  [. j  /  i ]. et )
bnj1030.18  |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i
)  ->  et' ) )
bnj1030.19  |-  ( ph0  <->  (
i  e.  n  /\  si 
/\  f  e.  K  /\  i  e.  dom  f ) )
Assertion
Ref Expression
bnj1030  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Distinct variable groups:    A, f,
i, j, n, y   
z, A, f, i, n    B, f, i, n, y    z, B    D, i, j    R, f, i, j, n, y    z, R    f, X, i, n, y    z, X    ch, j    et, j    ta, f,
i, j, n    th, f,
i, j, n    ph, i    ta, z    th, z
Allowed substitution hints:    ph( y, z, f, j, n)    ps( y, z, f, i, j, n)    ch( y, z, f, i, n)    th( y)    ta( y)    et( y, z, f, i, n)    ze( y,
z, f, i, j, n)    si( y, z, f, i, j, n)    rh( y, z, f, i, j, n)    B( j)    D( y, z, f, n)    K( y, z, f, i, j, n)    X( j)    ph'( y, z, f, i, j, n)    ps'( y, z, f, i, j, n)    ch'( y, z, f, i, j, n)    th'( y, z, f, i, j, n)    ta'( y, z, f, i, j, n)    et'( y, z, f, i, j, n)    ze'( y, z, f, i, j, n)    ph0( y, z, f, i, j, n)

Proof of Theorem bnj1030
StepHypRef Expression
1 bnj1030.1 . 2  |-  ( ph  <->  ( f `  (/) )  = 
pred ( X ,  A ,  R )
)
2 bnj1030.2 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj1030.3 . 2  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
4 bnj1030.4 . 2  |-  ( th  <->  ( R  FrSe  A  /\  X  e.  A )
)
5 bnj1030.5 . 2  |-  ( ta  <->  ( B  e.  _V  /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B
) )
6 bnj1030.6 . 2  |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i ) ) )
7 bnj1030.7 . 2  |-  D  =  ( om  \  { (/)
} )
8 bnj1030.8 . 2  |-  K  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
9 19.23vv 1833 . . . . 5  |-  ( A. n A. i ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )  <-> 
( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
) )
109albii 1553 . . . 4  |-  ( A. f A. n A. i
( ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)  <->  A. f ( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) )
11 19.23v 1832 . . . 4  |-  ( A. f ( E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)  <->  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) )
1210, 11bitri 240 . . 3  |-  ( A. f A. n A. i
( ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)  <->  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B ) )
13 bnj1030.9 . . . . 5  |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  (
f `  i )  C_  B ) )
147bnj1071 29007 . . . . . . . 8  |-  ( n  e.  D  ->  _E  Fr  n )
153, 14bnj769 28792 . . . . . . 7  |-  ( ch 
->  _E  Fr  n )
1615bnj707 28784 . . . . . 6  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  _E  Fr  n )
17 bnj1030.10 . . . . . . 7  |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  [. j  /  i ]. et ) )
18 bnj1030.17 . . . . . . 7  |-  ( et'  <->  [. j  /  i ]. et )
19 bnj1030.18 . . . . . . 7  |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i
)  ->  et' ) )
20 bnj1030.19 . . . . . . 7  |-  ( ph0  <->  (
i  e.  n  /\  si 
/\  f  e.  K  /\  i  e.  dom  f ) )
212, 8, 13, 18bnj1123 29016 . . . . . . . . . 10  |-  ( et'  <->  (
( f  e.  K  /\  j  e.  dom  f )  ->  (
f `  j )  C_  B ) )
222, 3, 5, 7, 19, 20, 21bnj1118 29014 . . . . . . . . 9  |-  E. j
( ( i  =/=  (/)  /\  ( ( th 
/\  ta  /\  ch )  /\  ph0 ) )  -> 
( f `  i
)  C_  B )
231, 3, 5bnj1097 29011 . . . . . . . . 9  |-  ( ( i  =  (/)  /\  (
( th  /\  ta  /\ 
ch )  /\  ph0 )
)  ->  ( f `  i )  C_  B
)
2422, 23bnj1109 28818 . . . . . . . 8  |-  E. j
( ( ( th 
/\  ta  /\  ch )  /\  ph0 )  ->  (
f `  i )  C_  B )
2524, 2, 3bnj1093 29010 . . . . . . 7  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i E. j (
ph0  ->  ( f `  i )  C_  B
) )
2613, 17, 18, 19, 20, 25bnj1090 29009 . . . . . 6  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  ( rh  ->  et ) )
27 vex 2791 . . . . . . 7  |-  n  e. 
_V
2827, 17bnj110 28890 . . . . . 6  |-  ( (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et )
)  ->  A. i  e.  n  et )
2916, 26, 28syl2anc 642 . . . . 5  |-  ( ( th  /\  ta  /\  ch  /\  ze )  ->  A. i  e.  n  et )
304, 5, 3, 6, 13, 29, 8bnj1121 29015 . . . 4  |-  ( ( th  /\  ta  /\  ch  /\  ze )  -> 
z  e.  B )
3130gen2 1534 . . 3  |-  A. n A. i ( ( th 
/\  ta  /\  ch  /\  ze )  ->  z  e.  B )
3212, 31mpgbi 1536 . 2  |-  ( E. f E. n E. i ( th  /\  ta  /\  ch  /\  ze )  ->  z  e.  B
)
331, 2, 3, 4, 5, 6, 7, 8, 32bnj1034 29000 1  |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   [.wsbc 2991    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   U_ciun 3905   class class class wbr 4023    _E cep 4303    Fr wfr 4349   suc csuc 4394   omcom 4656   dom cdm 4689    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719    TrFow-bnj19 28721
This theorem is referenced by:  bnj1124  29018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-iota 5219  df-fn 5258  df-fv 5263  df-bnj17 28712  df-bnj18 28720  df-bnj19 28722
  Copyright terms: Public domain W3C validator