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Theorem bnj966 28976
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
bnj966.3  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
bnj966.10  |-  D  =  ( om  \  { (/)
} )
bnj966.12  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
bnj966.13  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj966.44  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
bnj966.53  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  G  Fn  p )
Assertion
Ref Expression
bnj966  |-  ( ( ( 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 ) )
Distinct variable groups:    y, f    y, i    y, m    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 bnj966
StepHypRef Expression
1 bnj966.53 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  G  Fn  p )
21bnj930 28801 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  Fun  G )
323adant3 975 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  Fun  G )
4 opex 4237 . . . . . . 7  |-  <. n ,  C >.  e.  _V
54snid 3667 . . . . . 6  |-  <. n ,  C >.  e.  { <. n ,  C >. }
6 elun2 3343 . . . . . 6  |-  ( <.
n ,  C >.  e. 
{ <. n ,  C >. }  ->  <. n ,  C >.  e.  (
f  u.  { <. n ,  C >. } ) )
75, 6ax-mp 8 . . . . 5  |-  <. n ,  C >.  e.  (
f  u.  { <. n ,  C >. } )
8 bnj966.13 . . . . 5  |-  G  =  ( f  u.  { <. n ,  C >. } )
97, 8eleqtrri 2356 . . . 4  |-  <. n ,  C >.  e.  G
10 funopfv 5562 . . . 4  |-  ( Fun 
G  ->  ( <. n ,  C >.  e.  G  ->  ( G `  n
)  =  C ) )
113, 9, 10ee10 1366 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
( G `  n
)  =  C )
12 simp22 989 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  n  =  suc  m )
13 simp33 993 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  n  =  suc  i )
14 bnj551 28771 . . . . 5  |-  ( ( n  =  suc  m  /\  n  =  suc  i )  ->  m  =  i )
1512, 13, 14syl2anc 642 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  m  =  i )
16 suceq 4457 . . . . . . . 8  |-  ( m  =  i  ->  suc  m  =  suc  i )
1716eqeq2d 2294 . . . . . . 7  |-  ( m  =  i  ->  (
n  =  suc  m  <->  n  =  suc  i ) )
1817biimpac 472 . . . . . 6  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  n  =  suc  i )
1918fveq2d 5529 . . . . 5  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  ( G `  n )  =  ( G `  suc  i
) )
20 bnj966.12 . . . . . . 7  |-  C  = 
U_ y  e.  ( f `  m ) 
pred ( y ,  A ,  R )
21 fveq2 5525 . . . . . . . 8  |-  ( m  =  i  ->  (
f `  m )  =  ( f `  i ) )
2221bnj1113 28817 . . . . . . 7  |-  ( m  =  i  ->  U_ y  e.  ( f `  m
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2320, 22syl5eq 2327 . . . . . 6  |-  ( m  =  i  ->  C  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )
2423adantl 452 . . . . 5  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  C  =  U_ y  e.  ( f `
 i )  pred ( y ,  A ,  R ) )
2519, 24eqeq12d 2297 . . . 4  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  ( ( G `  n )  =  C  <->  ( G `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2612, 15, 25syl2anc 642 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
( ( G `  n )  =  C  <-> 
( G `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2711, 26mpbid 201 . 2  |-  ( ( ( 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.  ( f `  i )  pred (
y ,  A ,  R ) )
28 bnj966.44 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )
29283adant3 975 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  C  e.  _V )
30 bnj966.3 . . . . . . . 8  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
3130bnj1235 28837 . . . . . . 7  |-  ( ch 
->  f  Fn  n
)
32313ad2ant1 976 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
f  Fn  n )
33323ad2ant2 977 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
f  Fn  n )
34 simp23 990 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  p  =  suc  n )
3529, 33, 34, 13bnj951 28807 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i ) )
36 bnj966.10 . . . . . . . . 9  |-  D  =  ( om  \  { (/)
} )
3736bnj923 28798 . . . . . . . 8  |-  ( n  e.  D  ->  n  e.  om )
3830, 37bnj769 28792 . . . . . . 7  |-  ( ch 
->  n  e.  om )
39383ad2ant1 976 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  om )
40 simp3 957 . . . . . 6  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i )  ->  n  =  suc  i )
4139, 40bnj240 28724 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
( n  e.  om  /\  n  =  suc  i
) )
42 vex 2791 . . . . . . 7  |-  i  e. 
_V
4342bnj216 28760 . . . . . 6  |-  ( n  =  suc  i  -> 
i  e.  n )
4443adantl 452 . . . . 5  |-  ( ( n  e.  om  /\  n  =  suc  i )  ->  i  e.  n
)
4541, 44syl 15 . . . 4  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
i  e.  n )
46 bnj658 28780 . . . . . . 7  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i )  ->  ( C  e. 
_V  /\  f  Fn  n  /\  p  =  suc  n ) )
4746anim1i 551 . . . . . 6  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i )  /\  i  e.  n
)  ->  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n )  /\  i  e.  n )
)
48 df-bnj17 28712 . . . . . 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
) )
4947, 48sylibr 203 . . . . 5  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i )  /\  i  e.  n
)  ->  ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n ) )
508bnj945 28805 . . . . 5  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n )  ->  ( G `  i
)  =  ( f `
 i ) )
5149, 50syl 15 . . . 4  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i )  /\  i  e.  n
)  ->  ( G `  i )  =  ( f `  i ) )
5235, 45, 51syl2anc 642 . . 3  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  (
i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  -> 
( G `  i
)  =  ( f `
 i ) )
5320, 8bnj958 28972 . . . . 5  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
5453bnj956 28808 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
5554eqeq2d 2294 . . 3  |-  ( ( G `  i )  =  ( f `  i )  ->  (
( G `  suc  i )  =  U_ y  e.  ( G `  i )  pred (
y ,  A ,  R )  <->  ( G `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
5652, 55syl 15 . 2  |-  ( ( ( 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 )  <->  ( G `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
5727, 56mpbird 223 1  |-  ( ( ( 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 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150   (/)c0 3455   {csn 3640   <.cop 3643   U_ciun 3905   suc csuc 4394   omcom 4656   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj910  28980
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  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-eprel 4305  df-id 4309  df-fr 4352  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-bnj17 28712
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