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Theorem bnj1326 31874
Description: Technical lemma for bnj60 31910. 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 2497 . . . 4  |-  ( q  =  h  ->  (
q  e.  C  <->  h  e.  C ) )
213anbi3d 1295 . . 3  |-  ( q  =  h  ->  (
( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
) ) )
3 dmeq 5032 . . . . . . 7  |-  ( q  =  h  ->  dom  q  =  dom  h )
43ineq2d 3545 . . . . . 6  |-  ( q  =  h  ->  ( dom  g  i^i  dom  q
)  =  ( dom  g  i^i  dom  h
) )
54reseq2d 5102 . . . . 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 5099 . . . . 5  |-  ( g  |`  D )  =  ( g  |`  ( dom  g  i^i  dom  h )
)
85, 7syl6eqr 2487 . . . 4  |-  ( q  =  h  ->  (
g  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  D ) )
94reseq2d 5102 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  h ) ) )
10 reseq1 5096 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  h ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
119, 10eqtrd 2469 . . . . 5  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
126reseq2i 5099 . . . . 5  |-  ( h  |`  D )  =  ( h  |`  ( dom  g  i^i  dom  h )
)
1311, 12syl6eqr 2487 . . . 4  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  D ) )
148, 13eqeq12d 2451 . . 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 2497 . . . . 5  |-  ( p  =  g  ->  (
p  e.  C  <->  g  e.  C ) )
17163anbi2d 1294 . . . 4  |-  ( p  =  g  ->  (
( R  FrSe  A  /\  p  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
) ) )
18 dmeq 5032 . . . . . . . 8  |-  ( p  =  g  ->  dom  p  =  dom  g )
1918ineq1d 3544 . . . . . . 7  |-  ( p  =  g  ->  ( dom  p  i^i  dom  q
)  =  ( dom  g  i^i  dom  q
) )
2019reseq2d 5102 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( p  |`  ( dom  g  i^i  dom  q ) ) )
21 reseq1 5096 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2220, 21eqtrd 2469 . . . . 5  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2319reseq2d 5102 . . . . 5  |-  ( p  =  g  ->  (
q  |`  ( dom  p  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  q ) ) )
2422, 23eqeq12d 2451 . . . 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 2437 . . . 4  |-  ( dom  p  i^i  dom  q
)  =  ( dom  p  i^i  dom  q
)
3026, 27, 28, 29bnj1311 31872 . . 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 1958 . 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 1958 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 965    = wceq 1369    e. wcel 1756   {cab 2423   A.wral 2709   E.wrex 2710    i^i cin 3320    C_ wss 3321   <.cop 3876   dom cdm 4832    |` cres 4834    Fn wfn 5406   ` cfv 5411    predc-bnj14 31533    FrSe w-bnj15 31537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-reg 7799  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-om 6472  df-1o 6912  df-bnj17 31532  df-bnj14 31534  df-bnj13 31536  df-bnj15 31538  df-bnj18 31540  df-bnj19 31542
This theorem is referenced by:  bnj1321  31875  bnj1384  31880
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