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Theorem bnj1326 29056
Description: Technical lemma for bnj60 29092. 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 2343 . . . 4  |-  ( q  =  h  ->  (
q  e.  C  <->  h  e.  C ) )
213anbi3d 1258 . . 3  |-  ( q  =  h  ->  (
( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
) ) )
3 dmeq 4879 . . . . . . 7  |-  ( q  =  h  ->  dom  q  =  dom  h )
43ineq2d 3370 . . . . . 6  |-  ( q  =  h  ->  ( dom  g  i^i  dom  q
)  =  ( dom  g  i^i  dom  h
) )
54reseq2d 4955 . . . . 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 4952 . . . . 5  |-  ( g  |`  D )  =  ( g  |`  ( dom  g  i^i  dom  h )
)
85, 7syl6eqr 2333 . . . 4  |-  ( q  =  h  ->  (
g  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  D ) )
94reseq2d 4955 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  h ) ) )
10 reseq1 4949 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  h ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
119, 10eqtrd 2315 . . . . 5  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
126reseq2i 4952 . . . . 5  |-  ( h  |`  D )  =  ( h  |`  ( dom  g  i^i  dom  h )
)
1311, 12syl6eqr 2333 . . . 4  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  D ) )
148, 13eqeq12d 2297 . . 3  |-  ( q  =  h  ->  (
( g  |`  ( dom  g  i^i  dom  q
) )  =  ( q  |`  ( dom  g  i^i  dom  q )
)  <->  ( g  |`  D )  =  ( h  |`  D )
) )
152, 14imbi12d 311 . 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 2343 . . . . 5  |-  ( p  =  g  ->  (
p  e.  C  <->  g  e.  C ) )
17163anbi2d 1257 . . . 4  |-  ( p  =  g  ->  (
( R  FrSe  A  /\  p  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
) ) )
18 dmeq 4879 . . . . . . . 8  |-  ( p  =  g  ->  dom  p  =  dom  g )
1918ineq1d 3369 . . . . . . 7  |-  ( p  =  g  ->  ( dom  p  i^i  dom  q
)  =  ( dom  g  i^i  dom  q
) )
2019reseq2d 4955 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( p  |`  ( dom  g  i^i  dom  q ) ) )
21 reseq1 4949 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2220, 21eqtrd 2315 . . . . 5  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2319reseq2d 4955 . . . . 5  |-  ( p  =  g  ->  (
q  |`  ( dom  p  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  q ) ) )
2422, 23eqeq12d 2297 . . . 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 311 . . 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 2283 . . . 4  |-  ( dom  p  i^i  dom  q
)  =  ( dom  p  i^i  dom  q
)
3026, 27, 28, 29bnj1311 29054 . . 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 1953 . 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 1953 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   <.cop 3643   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj1321  29057  bnj1384  29062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718  df-bnj18 28720  df-bnj19 28722
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