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Theorem bnj1326 29395
Description: Technical lemma for bnj60 29431. 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 2496 . . . 4  |-  ( q  =  h  ->  (
q  e.  C  <->  h  e.  C ) )
213anbi3d 1260 . . 3  |-  ( q  =  h  ->  (
( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
) ) )
3 dmeq 5070 . . . . . . 7  |-  ( q  =  h  ->  dom  q  =  dom  h )
43ineq2d 3542 . . . . . 6  |-  ( q  =  h  ->  ( dom  g  i^i  dom  q
)  =  ( dom  g  i^i  dom  h
) )
54reseq2d 5146 . . . . 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 5143 . . . . 5  |-  ( g  |`  D )  =  ( g  |`  ( dom  g  i^i  dom  h )
)
85, 7syl6eqr 2486 . . . 4  |-  ( q  =  h  ->  (
g  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  D ) )
94reseq2d 5146 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  h ) ) )
10 reseq1 5140 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  h ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
119, 10eqtrd 2468 . . . . 5  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
126reseq2i 5143 . . . . 5  |-  ( h  |`  D )  =  ( h  |`  ( dom  g  i^i  dom  h )
)
1311, 12syl6eqr 2486 . . . 4  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  D ) )
148, 13eqeq12d 2450 . . 3  |-  ( q  =  h  ->  (
( g  |`  ( dom  g  i^i  dom  q
) )  =  ( q  |`  ( dom  g  i^i  dom  q )
)  <->  ( g  |`  D )  =  ( h  |`  D )
) )
152, 14imbi12d 312 . 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 2496 . . . . 5  |-  ( p  =  g  ->  (
p  e.  C  <->  g  e.  C ) )
17163anbi2d 1259 . . . 4  |-  ( p  =  g  ->  (
( R  FrSe  A  /\  p  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
) ) )
18 dmeq 5070 . . . . . . . 8  |-  ( p  =  g  ->  dom  p  =  dom  g )
1918ineq1d 3541 . . . . . . 7  |-  ( p  =  g  ->  ( dom  p  i^i  dom  q
)  =  ( dom  g  i^i  dom  q
) )
2019reseq2d 5146 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( p  |`  ( dom  g  i^i  dom  q ) ) )
21 reseq1 5140 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2220, 21eqtrd 2468 . . . . 5  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2319reseq2d 5146 . . . . 5  |-  ( p  =  g  ->  (
q  |`  ( dom  p  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  q ) ) )
2422, 23eqeq12d 2450 . . . 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 312 . . 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 2436 . . . 4  |-  ( dom  p  i^i  dom  q
)  =  ( dom  p  i^i  dom  q
)
3026, 27, 28, 29bnj1311 29393 . . 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 1969 . 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 1969 1  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  ( g  |`  D )  =  ( h  |`  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   <.cop 3817   dom cdm 4878    |` cres 4880    Fn wfn 5449   ` cfv 5454    predc-bnj14 29052    FrSe w-bnj15 29056
This theorem is referenced by:  bnj1321  29396  bnj1384  29401
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-bnj17 29051  df-bnj14 29053  df-bnj13 29055  df-bnj15 29057  df-bnj18 29059  df-bnj19 29061
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