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Theorem bnj966 31934
Description: Technical lemma for bnj69 31998. 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 31760 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  Fun  G )
323adant3 1008 . . . 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 4554 . . . . . . 7  |-  <. n ,  C >.  e.  _V
54snid 3903 . . . . . 6  |-  <. n ,  C >.  e.  { <. n ,  C >. }
6 elun2 3522 . . . . . 6  |-  ( <.
n ,  C >.  e. 
{ <. n ,  C >. }  ->  <. n ,  C >.  e.  (
f  u.  { <. n ,  C >. } ) )
75, 6ax-mp 5 . . . . 5  |-  <. n ,  C >.  e.  (
f  u.  { <. n ,  C >. } )
8 bnj966.13 . . . . 5  |-  G  =  ( f  u.  { <. n ,  C >. } )
97, 8eleqtrri 2514 . . . 4  |-  <. n ,  C >.  e.  G
10 funopfv 5729 . . . 4  |-  ( Fun 
G  ->  ( <. n ,  C >.  e.  G  ->  ( G `  n
)  =  C ) )
113, 9, 10mpisyl 18 . . 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 1022 . . . 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 1026 . . . . 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 31731 . . . . 5  |-  ( ( n  =  suc  m  /\  n  =  suc  i )  ->  m  =  i )
1512, 13, 14syl2anc 661 . . . 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 4782 . . . . . . . 8  |-  ( m  =  i  ->  suc  m  =  suc  i )
1716eqeq2d 2452 . . . . . . 7  |-  ( m  =  i  ->  (
n  =  suc  m  <->  n  =  suc  i ) )
1817biimpac 486 . . . . . 6  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  n  =  suc  i )
1918fveq2d 5693 . . . . 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 5689 . . . . . . . 8  |-  ( m  =  i  ->  (
f `  m )  =  ( f `  i ) )
2221bnj1113 31776 . . . . . . 7  |-  ( m  =  i  ->  U_ y  e.  ( f `  m
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2320, 22syl5eq 2485 . . . . . 6  |-  ( m  =  i  ->  C  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )
2423adantl 466 . . . . 5  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  C  =  U_ y  e.  ( f `
 i )  pred ( y ,  A ,  R ) )
2519, 24eqeq12d 2455 . . . 4  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  ( ( G `  n )  =  C  <->  ( G `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2612, 15, 25syl2anc 661 . . 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 210 . 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 1008 . . . . 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 31795 . . . . . . 7  |-  ( ch 
->  f  Fn  n
)
32313ad2ant1 1009 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
f  Fn  n )
33323ad2ant2 1010 . . . . 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 1023 . . . . 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 31766 . . . 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 31758 . . . . . . . 8  |-  ( n  e.  D  ->  n  e.  om )
3830, 37bnj769 31752 . . . . . . 7  |-  ( ch 
->  n  e.  om )
39383ad2ant1 1009 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  om )
40 simp3 990 . . . . . 6  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i )  ->  n  =  suc  i )
4139, 40bnj240 31684 . . . . 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 2973 . . . . . . 7  |-  i  e. 
_V
4342bnj216 31720 . . . . . 6  |-  ( n  =  suc  i  -> 
i  e.  n )
4443adantl 466 . . . . 5  |-  ( ( n  e.  om  /\  n  =  suc  i )  ->  i  e.  n
)
4541, 44syl 16 . . . 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 31740 . . . . . . 7  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i )  ->  ( C  e. 
_V  /\  f  Fn  n  /\  p  =  suc  n ) )
4746anim1i 568 . . . . . 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 31672 . . . . . 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 212 . . . . 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 31764 . . . . 5  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n )  ->  ( G `  i
)  =  ( f `
 i ) )
5149, 50syl 16 . . . 4  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i )  /\  i  e.  n
)  ->  ( G `  i )  =  ( f `  i ) )
5235, 45, 51syl2anc 661 . . 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 31930 . . . . 5  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
5453bnj956 31767 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
5554eqeq2d 2452 . . 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 16 . 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 232 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2970    \ cdif 3323    u. cun 3324   (/)c0 3635   {csn 3875   <.cop 3881   U_ciun 4169   suc csuc 4719   Fun wfun 5410    Fn wfn 5411   ` cfv 5416   omcom 6474    /\ w-bnj17 31671    predc-bnj14 31673    FrSe w-bnj15 31677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-un 6370  ax-reg 7805
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-eprel 4630  df-id 4634  df-fr 4677  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-res 4850  df-iota 5379  df-fun 5418  df-fn 5419  df-fv 5424  df-bnj17 31672
This theorem is referenced by:  bnj910  31938
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