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Theorem bnj966 29757
Description: Technical lemma for bnj69 29821. 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 29583 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  Fun  G )
323adant3 1026 . . . 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 4683 . . . . . . 7  |-  <. n ,  C >.  e.  _V
54snid 4025 . . . . . 6  |-  <. n ,  C >.  e.  { <. n ,  C >. }
6 elun2 3635 . . . . . 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 2510 . . . 4  |-  <. n ,  C >.  e.  G
10 funopfv 5918 . . . 4  |-  ( Fun 
G  ->  ( <. n ,  C >.  e.  G  ->  ( G `  n
)  =  C ) )
113, 9, 10mpisyl 22 . . 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 1040 . . . 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 1044 . . . . 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 29554 . . . . 5  |-  ( ( n  =  suc  m  /\  n  =  suc  i )  ->  m  =  i )
1512, 13, 14syl2anc 666 . . . 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 5505 . . . . . . . 8  |-  ( m  =  i  ->  suc  m  =  suc  i )
1716eqeq2d 2437 . . . . . . 7  |-  ( m  =  i  ->  (
n  =  suc  m  <->  n  =  suc  i ) )
1817biimpac 489 . . . . . 6  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  n  =  suc  i )
1918fveq2d 5883 . . . . 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 5879 . . . . . . . 8  |-  ( m  =  i  ->  (
f `  m )  =  ( f `  i ) )
2221bnj1113 29599 . . . . . . 7  |-  ( m  =  i  ->  U_ y  e.  ( f `  m
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2320, 22syl5eq 2476 . . . . . 6  |-  ( m  =  i  ->  C  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )
2423adantl 468 . . . . 5  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  C  =  U_ y  e.  ( f `
 i )  pred ( y ,  A ,  R ) )
2519, 24eqeq12d 2445 . . . 4  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  ( ( G `  n )  =  C  <->  ( G `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2612, 15, 25syl2anc 666 . . 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 214 . 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 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 ) )  ->  C  e.  _V )
30 bnj966.3 . . . . . . . 8  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
3130bnj1235 29618 . . . . . . 7  |-  ( ch 
->  f  Fn  n
)
32313ad2ant1 1027 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
f  Fn  n )
33323ad2ant2 1028 . . . . 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 1041 . . . . 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 29589 . . . 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 29581 . . . . . . . 8  |-  ( n  e.  D  ->  n  e.  om )
3830, 37bnj769 29575 . . . . . . 7  |-  ( ch 
->  n  e.  om )
39383ad2ant1 1027 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  om )
40 simp3 1008 . . . . . 6  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i )  ->  n  =  suc  i )
4139, 40bnj240 29506 . . . . 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 3085 . . . . . . 7  |-  i  e. 
_V
4342bnj216 29542 . . . . . 6  |-  ( n  =  suc  i  -> 
i  e.  n )
4443adantl 468 . . . . 5  |-  ( ( n  e.  om  /\  n  =  suc  i )  ->  i  e.  n
)
4541, 44syl 17 . . . 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 29563 . . . . . . 7  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i )  ->  ( C  e. 
_V  /\  f  Fn  n  /\  p  =  suc  n ) )
4746anim1i 571 . . . . . 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 29494 . . . . . 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 216 . . . . 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 29587 . . . . 5  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  i  e.  n )  ->  ( G `  i
)  =  ( f `
 i ) )
5149, 50syl 17 . . . 4  |-  ( ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i )  /\  i  e.  n
)  ->  ( G `  i )  =  ( f `  i ) )
5235, 45, 51syl2anc 666 . . 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 29753 . . . . 5  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
5453bnj956 29590 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
5554eqeq2d 2437 . . 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 17 . 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 236 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   _Vcvv 3082    \ cdif 3434    u. cun 3435   (/)c0 3762   {csn 3997   <.cop 4003   U_ciun 4297   suc csuc 5442   Fun wfun 5593    Fn wfn 5594   ` cfv 5599   omcom 6704    /\ w-bnj17 29493    predc-bnj14 29495    FrSe w-bnj15 29499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658  ax-un 6595  ax-reg 8111
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-eprel 4762  df-id 4766  df-fr 4810  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-res 4863  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-fv 5607  df-bnj17 29494
This theorem is referenced by:  bnj910  29761
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