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Theorem bnj967 33100
Description: Technical lemma for bnj69 33163. 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 32960 . . . . . . . 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 32931 . . . . 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 32923 . . . . . . . . 9  |-  ( n  e.  D  ->  n  e.  om )
133, 12bnj769 32917 . . . . . . . 8  |-  ( ch 
->  n  e.  om )
14133ad2ant1 1017 . . . . . . 7  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  om )
1514, 8bnj240 32849 . . . . . 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 6692 . . . . . . . 8  |-  ( n  e.  om  ->  Ord  n )
17 ordtr 4892 . . . . . . . 8  |-  ( Ord  n  ->  Tr  n
)
1816, 17syl 16 . . . . . . 7  |-  ( n  e.  om  ->  Tr  n )
19 trsuc 4962 . . . . . . 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 32905 . . . . . . 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 32837 . . . . . 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 32929 . . . 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 32929 . . . 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 32965 . . . . 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 32931 . 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 33095 . . 3  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
4138, 40bnj953 33094 . 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 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    \ cdif 3473    u. cun 3474   (/)c0 3785   {csn 4027   <.cop 4033   U_ciun 4325   Tr wtr 4540   Ord word 4877   suc csuc 4880    Fn wfn 5583   ` cfv 5588   omcom 6684    /\ w-bnj17 32836    predc-bnj14 32838    FrSe w-bnj15 32842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576  ax-reg 8018
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-om 6685  df-bnj17 32837
This theorem is referenced by:  bnj910  33103
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