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Theorem bnj966 32956
Description: Technical lemma for bnj69 33020. 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 32782 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  Fun  G )
323adant3 1011 . . . 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 4704 . . . . . . 7  |-  <. n ,  C >.  e.  _V
54snid 4048 . . . . . 6  |-  <. n ,  C >.  e.  { <. n ,  C >. }
6 elun2 3665 . . . . . 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 2547 . . . 4  |-  <. n ,  C >.  e.  G
10 funopfv 5898 . . . 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 1025 . . . 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 1029 . . . . 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 32753 . . . . 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 4936 . . . . . . . 8  |-  ( m  =  i  ->  suc  m  =  suc  i )
1716eqeq2d 2474 . . . . . . 7  |-  ( m  =  i  ->  (
n  =  suc  m  <->  n  =  suc  i ) )
1817biimpac 486 . . . . . 6  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  n  =  suc  i )
1918fveq2d 5861 . . . . 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 5857 . . . . . . . 8  |-  ( m  =  i  ->  (
f `  m )  =  ( f `  i ) )
2221bnj1113 32798 . . . . . . 7  |-  ( m  =  i  ->  U_ y  e.  ( f `  m
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2320, 22syl5eq 2513 . . . . . 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 2482 . . . 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 1011 . . . . 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 32817 . . . . . . 7  |-  ( ch 
->  f  Fn  n
)
32313ad2ant1 1012 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
f  Fn  n )
33323ad2ant2 1013 . . . . 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 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 ) )  ->  p  =  suc  n )
3529, 33, 34, 13bnj951 32788 . . . 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 32780 . . . . . . . 8  |-  ( n  e.  D  ->  n  e.  om )
3830, 37bnj769 32774 . . . . . . 7  |-  ( ch 
->  n  e.  om )
39383ad2ant1 1012 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  om )
40 simp3 993 . . . . . 6  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i )  ->  n  =  suc  i )
4139, 40bnj240 32706 . . . . 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 3109 . . . . . . 7  |-  i  e. 
_V
4342bnj216 32742 . . . . . 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 32762 . . . . . . 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 32694 . . . . . 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 32786 . . . . 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 32952 . . . . 5  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
5453bnj956 32789 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
5554eqeq2d 2474 . . 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 968    = wceq 1374    e. wcel 1762   _Vcvv 3106    \ cdif 3466    u. cun 3467   (/)c0 3778   {csn 4020   <.cop 4026   U_ciun 4318   suc csuc 4873   Fun wfun 5573    Fn wfn 5574   ` cfv 5579   omcom 6671    /\ w-bnj17 32693    predc-bnj14 32695    FrSe w-bnj15 32699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567  ax-reg 8007
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-eprel 4784  df-id 4788  df-fr 4831  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587  df-bnj17 32694
This theorem is referenced by:  bnj910  32960
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