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Theorem bnj1326 33810
Description: Technical lemma for bnj60 33846. 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
bnj1326.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1326.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1326.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1326.4  |-  D  =  ( dom  g  i^i 
dom  h )
Assertion
Ref Expression
bnj1326  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  ( g  |`  D )  =  ( h  |`  D ) )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f    R, d, f, x
Allowed substitution hints:    A( g, h)    B( x, g, h, d)    C( x, f, g, h, d)    D( x, f, g, h, d)    R( g, h)    G( x, g, h)    Y( x, f, g, h, d)

Proof of Theorem bnj1326
Dummy variables  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2513 . . . 4  |-  ( q  =  h  ->  (
q  e.  C  <->  h  e.  C ) )
213anbi3d 1304 . . 3  |-  ( q  =  h  ->  (
( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
) ) )
3 dmeq 5190 . . . . . . 7  |-  ( q  =  h  ->  dom  q  =  dom  h )
43ineq2d 3683 . . . . . 6  |-  ( q  =  h  ->  ( dom  g  i^i  dom  q
)  =  ( dom  g  i^i  dom  h
) )
54reseq2d 5260 . . . . 5  |-  ( q  =  h  ->  (
g  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  h ) ) )
6 bnj1326.4 . . . . . 6  |-  D  =  ( dom  g  i^i 
dom  h )
76reseq2i 5257 . . . . 5  |-  ( g  |`  D )  =  ( g  |`  ( dom  g  i^i  dom  h )
)
85, 7syl6eqr 2500 . . . 4  |-  ( q  =  h  ->  (
g  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  D ) )
94reseq2d 5260 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  h ) ) )
10 reseq1 5254 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  h ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
119, 10eqtrd 2482 . . . . 5  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
126reseq2i 5257 . . . . 5  |-  ( h  |`  D )  =  ( h  |`  ( dom  g  i^i  dom  h )
)
1311, 12syl6eqr 2500 . . . 4  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  D ) )
148, 13eqeq12d 2463 . . 3  |-  ( q  =  h  ->  (
( g  |`  ( dom  g  i^i  dom  q
) )  =  ( q  |`  ( dom  g  i^i  dom  q )
)  <->  ( g  |`  D )  =  ( h  |`  D )
) )
152, 14imbi12d 320 . 2  |-  ( q  =  h  ->  (
( ( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
)  ->  ( g  |`  ( dom  g  i^i 
dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  q ) ) )  <-> 
( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
)  ->  ( g  |`  D )  =  ( h  |`  D )
) ) )
16 eleq1 2513 . . . . 5  |-  ( p  =  g  ->  (
p  e.  C  <->  g  e.  C ) )
17163anbi2d 1303 . . . 4  |-  ( p  =  g  ->  (
( R  FrSe  A  /\  p  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
) ) )
18 dmeq 5190 . . . . . . . 8  |-  ( p  =  g  ->  dom  p  =  dom  g )
1918ineq1d 3682 . . . . . . 7  |-  ( p  =  g  ->  ( dom  p  i^i  dom  q
)  =  ( dom  g  i^i  dom  q
) )
2019reseq2d 5260 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( p  |`  ( dom  g  i^i  dom  q ) ) )
21 reseq1 5254 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2220, 21eqtrd 2482 . . . . 5  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2319reseq2d 5260 . . . . 5  |-  ( p  =  g  ->  (
q  |`  ( dom  p  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  q ) ) )
2422, 23eqeq12d 2463 . . . 4  |-  ( p  =  g  ->  (
( p  |`  ( dom  p  i^i  dom  q
) )  =  ( q  |`  ( dom  p  i^i  dom  q )
)  <->  ( g  |`  ( dom  g  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  q
) ) ) )
2517, 24imbi12d 320 . . 3  |-  ( p  =  g  ->  (
( ( R  FrSe  A  /\  p  e.  C  /\  q  e.  C
)  ->  ( p  |`  ( dom  p  i^i 
dom  q ) )  =  ( q  |`  ( dom  p  i^i  dom  q ) ) )  <-> 
( ( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
)  ->  ( g  |`  ( dom  g  i^i 
dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  q ) ) ) ) )
26 bnj1326.1 . . . 4  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
27 bnj1326.2 . . . 4  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
28 bnj1326.3 . . . 4  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
29 eqid 2441 . . . 4  |-  ( dom  p  i^i  dom  q
)  =  ( dom  p  i^i  dom  q
)
3026, 27, 28, 29bnj1311 33808 . . 3  |-  ( ( R  FrSe  A  /\  p  e.  C  /\  q  e.  C )  ->  ( p  |`  ( dom  p  i^i  dom  q
) )  =  ( q  |`  ( dom  p  i^i  dom  q )
) )
3125, 30chvarv 1998 . 2  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  q  e.  C )  ->  ( g  |`  ( dom  g  i^i  dom  q
) )  =  ( q  |`  ( dom  g  i^i  dom  q )
) )
3215, 31chvarv 1998 1  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  ( g  |`  D )  =  ( h  |`  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   {cab 2426   A.wral 2791   E.wrex 2792    i^i cin 3458    C_ wss 3459   <.cop 4017   dom cdm 4986    |` cres 4988    Fn wfn 5570   ` cfv 5575    predc-bnj14 33468    FrSe w-bnj15 33472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-reg 8018  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-om 6683  df-1o 7129  df-bnj17 33467  df-bnj14 33469  df-bnj13 33471  df-bnj15 33473  df-bnj18 33475  df-bnj19 33477
This theorem is referenced by:  bnj1321  33811  bnj1384  33816
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