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Theorem bnj967 34146
Description: Technical lemma for bnj69 34209. 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
bnj967.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj967.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj967.10  |-  D  =  ( om  \  { (/)
} )
bnj967.12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj967.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj967.44  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
Assertion
Ref Expression
bnj967  |-  ( ( ( 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 ) )
Distinct variable groups:    y, f    y, i    y, n
Allowed substitution hints:    ph( y, f, i, m, n, p)    ps( y, f, i, m, n, p)    ch( y,
f, i, m, n, p)    A( y, f, i, m, n, p)    C( y, f, i, m, n, p)    D( y, f, i, m, n, p)    R( y, f, i, m, n, p)    G( y, f, i, m, n, p)    X( y, f, i, m, n, p)

Proof of Theorem bnj967
StepHypRef Expression
1 bnj967.44 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
213adant3 1016 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  C  e.  _V )
3 bnj967.3 . . . . . . . . 9  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
43bnj1235 34006 . . . . . . . 8  |-  ( ch 
->  f  Fn  n
)
543ad2ant1 1017 . . . . . . 7  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
f  Fn  n )
653ad2ant2 1018 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  f  Fn  n )
7 simp23 1031 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  p  =  suc  n )
8 simp3 998 . . . . . . 7  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
)  ->  suc  i  e.  n )
983ad2ant3 1019 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  suc  i  e.  n )
102, 6, 7, 9bnj951 33977 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n )
)
11 bnj967.10 . . . . . . . . . 10  |-  D  =  ( om  \  { (/)
} )
1211bnj923 33969 . . . . . . . . 9  |-  ( n  e.  D  ->  n  e.  om )
133, 12bnj769 33963 . . . . . . . 8  |-  ( ch 
->  n  e.  om )
14133ad2ant1 1017 . . . . . . 7  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  om )
1514, 8bnj240 33894 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  (
n  e.  om  /\  suc  i  e.  n
) )
16 nnord 6707 . . . . . . . 8  |-  ( n  e.  om  ->  Ord  n )
17 ordtr 4901 . . . . . . . 8  |-  ( Ord  n  ->  Tr  n
)
1816, 17syl 16 . . . . . . 7  |-  ( n  e.  om  ->  Tr  n )
19 trsuc 4971 . . . . . . 7  |-  ( ( Tr  n  /\  suc  i  e.  n )  ->  i  e.  n )
2018, 19sylan 471 . . . . . 6  |-  ( ( n  e.  om  /\  suc  i  e.  n
)  ->  i  e.  n )
2115, 20syl 16 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  i  e.  n )
22 bnj658 33951 . . . . . . 7  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n
)  ->  ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n ) )
2322anim1i 568 . . . . . 6  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n
)  /\  i  e.  n )  ->  (
( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  /\  i  e.  n
) )
24 df-bnj17 33882 . . . . . 6  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n )  <->  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  /\  i  e.  n
) )
2523, 24sylibr 212 . . . . 5  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n
)  /\  i  e.  n )  ->  ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n ) )
2610, 21, 25syl2anc 661 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n ) )
27 bnj967.13 . . . . 5  |-  G  =  ( f  u.  { <. n ,  C >. } )
2827bnj945 33975 . . . 4  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n )  ->  ( G `  i
)  =  ( f `
 i ) )
2926, 28syl 16 . . 3  |-  ( ( ( 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 `  i )  =  ( f `  i ) )
3027bnj945 33975 . . . 4  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  suc  i  e.  n
)  ->  ( G `  suc  i )  =  ( f `  suc  i ) )
3110, 30syl 16 . . 3  |-  ( ( ( 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 )  =  ( f `  suc  i ) )
32 3simpb 994 . . . 4  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
)  ->  ( i  e.  om  /\  suc  i  e.  n ) )
33323ad2ant3 1019 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  (
i  e.  om  /\  suc  i  e.  n
) )
343bnj1254 34011 . . . . 5  |-  ( ch 
->  ps )
35343ad2ant1 1017 . . . 4  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  ps )
36353ad2ant2 1018 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n
) )  ->  ps )
3729, 31, 33, 36bnj951 33977 . 2  |-  ( ( ( 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 `  i
)  =  ( f `
 i )  /\  ( G `  suc  i
)  =  ( f `
 suc  i )  /\  ( i  e.  om  /\ 
suc  i  e.  n
)  /\  ps )
)
38 bnj967.2 . . 3  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
39 bnj967.12 . . . 4  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
4039, 27bnj958 34141 . . 3  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
4138, 40bnj953 34140 . 2  |-  ( ( ( 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 ) )
4237, 41syl 16 1  |-  ( ( ( 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 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    \ cdif 3468    u. cun 3469   (/)c0 3793   {csn 4032   <.cop 4038   U_ciun 4332   Tr wtr 4550   Ord word 4886   suc csuc 4889    Fn wfn 5589   ` cfv 5594   omcom 6699    /\ w-bnj17 33881    predc-bnj14 33883    FrSe w-bnj15 33887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591  ax-reg 8036
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-om 6700  df-bnj17 33882
This theorem is referenced by:  bnj910  34149
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