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Theorem bnj966 29252
Description: Technical lemma for bnj69 29316. 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 29077 . . . . 5  |-  ( ( ( R  FrSe  A  /\  X  e.  A
)  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  Fun  G )
323adant3 977 . . . 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 4419 . . . . . . 7  |-  <. n ,  C >.  e.  _V
54snid 3833 . . . . . 6  |-  <. n ,  C >.  e.  { <. n ,  C >. }
6 elun2 3507 . . . . . 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 2508 . . . 4  |-  <. n ,  C >.  e.  G
10 funopfv 5758 . . . 4  |-  ( Fun 
G  ->  ( <. n ,  C >.  e.  G  ->  ( G `  n
)  =  C ) )
113, 9, 10ee10 1385 . . 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 991 . . . 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 995 . . . . 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 29047 . . . . 5  |-  ( ( n  =  suc  m  /\  n  =  suc  i )  ->  m  =  i )
1512, 13, 14syl2anc 643 . . . 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 4638 . . . . . . . 8  |-  ( m  =  i  ->  suc  m  =  suc  i )
1716eqeq2d 2446 . . . . . . 7  |-  ( m  =  i  ->  (
n  =  suc  m  <->  n  =  suc  i ) )
1817biimpac 473 . . . . . 6  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  n  =  suc  i )
1918fveq2d 5724 . . . . 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 5720 . . . . . . . 8  |-  ( m  =  i  ->  (
f `  m )  =  ( f `  i ) )
2221bnj1113 29093 . . . . . . 7  |-  ( m  =  i  ->  U_ y  e.  ( f `  m
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
2320, 22syl5eq 2479 . . . . . 6  |-  ( m  =  i  ->  C  =  U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )
2423adantl 453 . . . . 5  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  C  =  U_ y  e.  ( f `
 i )  pred ( y ,  A ,  R ) )
2519, 24eqeq12d 2449 . . . 4  |-  ( ( n  =  suc  m  /\  m  =  i
)  ->  ( ( G `  n )  =  C  <->  ( G `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
2612, 15, 25syl2anc 643 . . 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 202 . 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 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 ) )  ->  C  e.  _V )
30 bnj966.3 . . . . . . . 8  |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph 
/\  ps ) )
3130bnj1235 29113 . . . . . . 7  |-  ( ch 
->  f  Fn  n
)
32313ad2ant1 978 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  -> 
f  Fn  n )
33323ad2ant2 979 . . . . 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 992 . . . . 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 29083 . . . 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 29074 . . . . . . . 8  |-  ( n  e.  D  ->  n  e.  om )
3830, 37bnj769 29068 . . . . . . 7  |-  ( ch 
->  n  e.  om )
39383ad2ant1 978 . . . . . 6  |-  ( ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  ->  n  e.  om )
40 simp3 959 . . . . . 6  |-  ( ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i )  ->  n  =  suc  i )
4139, 40bnj240 29000 . . . . 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 2951 . . . . . . 7  |-  i  e. 
_V
4342bnj216 29036 . . . . . 6  |-  ( n  =  suc  i  -> 
i  e.  n )
4443adantl 453 . . . . 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 29056 . . . . . . 7  |-  ( ( C  e.  _V  /\  f  Fn  n  /\  p  =  suc  n  /\  n  =  suc  i )  ->  ( C  e. 
_V  /\  f  Fn  n  /\  p  =  suc  n ) )
4746anim1i 552 . . . . . 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 28988 . . . . . 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 204 . . . . 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 29081 . . . . 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 643 . . 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 29248 . . . . 5  |-  ( ( G `  i )  =  ( f `  i )  ->  A. y
( G `  i
)  =  ( f `
 i ) )
5453bnj956 29084 . . . 4  |-  ( ( G `  i )  =  ( f `  i )  ->  U_ y  e.  ( G `  i
)  pred ( y ,  A ,  R )  =  U_ y  e.  ( f `  i
)  pred ( y ,  A ,  R ) )
5554eqeq2d 2446 . . 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 224 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310   (/)c0 3620   {csn 3806   <.cop 3809   U_ciun 4085   suc csuc 4575   omcom 4837   Fun wfun 5440    Fn wfn 5441   ` cfv 5446    /\ w-bnj17 28987    predc-bnj14 28989    FrSe w-bnj15 28993
This theorem is referenced by:  bnj910  29256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693  ax-reg 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-eprel 4486  df-id 4490  df-fr 4533  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-bnj17 28988
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