Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1326 Structured version   Unicode version

Theorem bnj1326 33562
Description: Technical lemma for bnj60 33598. 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 2539 . . . 4  |-  ( q  =  h  ->  (
q  e.  C  <->  h  e.  C ) )
213anbi3d 1305 . . 3  |-  ( q  =  h  ->  (
( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
) ) )
3 dmeq 5209 . . . . . . 7  |-  ( q  =  h  ->  dom  q  =  dom  h )
43ineq2d 3705 . . . . . 6  |-  ( q  =  h  ->  ( dom  g  i^i  dom  q
)  =  ( dom  g  i^i  dom  h
) )
54reseq2d 5279 . . . . 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 5276 . . . . 5  |-  ( g  |`  D )  =  ( g  |`  ( dom  g  i^i  dom  h )
)
85, 7syl6eqr 2526 . . . 4  |-  ( q  =  h  ->  (
g  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  D ) )
94reseq2d 5279 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  h ) ) )
10 reseq1 5273 . . . . . 6  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  h ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
119, 10eqtrd 2508 . . . . 5  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
126reseq2i 5276 . . . . 5  |-  ( h  |`  D )  =  ( h  |`  ( dom  g  i^i  dom  h )
)
1311, 12syl6eqr 2526 . . . 4  |-  ( q  =  h  ->  (
q  |`  ( dom  g  i^i  dom  q ) )  =  ( h  |`  D ) )
148, 13eqeq12d 2489 . . 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 2539 . . . . 5  |-  ( p  =  g  ->  (
p  e.  C  <->  g  e.  C ) )
17163anbi2d 1304 . . . 4  |-  ( p  =  g  ->  (
( R  FrSe  A  /\  p  e.  C  /\  q  e.  C
)  <->  ( R  FrSe  A  /\  g  e.  C  /\  q  e.  C
) ) )
18 dmeq 5209 . . . . . . . 8  |-  ( p  =  g  ->  dom  p  =  dom  g )
1918ineq1d 3704 . . . . . . 7  |-  ( p  =  g  ->  ( dom  p  i^i  dom  q
)  =  ( dom  g  i^i  dom  q
) )
2019reseq2d 5279 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( p  |`  ( dom  g  i^i  dom  q ) ) )
21 reseq1 5273 . . . . . 6  |-  ( p  =  g  ->  (
p  |`  ( dom  g  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2220, 21eqtrd 2508 . . . . 5  |-  ( p  =  g  ->  (
p  |`  ( dom  p  i^i  dom  q ) )  =  ( g  |`  ( dom  g  i^i  dom  q ) ) )
2319reseq2d 5279 . . . . 5  |-  ( p  =  g  ->  (
q  |`  ( dom  p  i^i  dom  q ) )  =  ( q  |`  ( dom  g  i^i  dom  q ) ) )
2422, 23eqeq12d 2489 . . . 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 2467 . . . 4  |-  ( dom  p  i^i  dom  q
)  =  ( dom  p  i^i  dom  q
)
3026, 27, 28, 29bnj1311 33560 . . 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 1983 . 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 1983 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 973    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817   E.wrex 2818    i^i cin 3480    C_ wss 3481   <.cop 4039   dom cdm 5005    |` cres 5007    Fn wfn 5589   ` cfv 5594    predc-bnj14 33221    FrSe w-bnj15 33225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-reg 8030  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-bnj17 33220  df-bnj14 33222  df-bnj13 33224  df-bnj15 33226  df-bnj18 33228  df-bnj19 33230
This theorem is referenced by:  bnj1321  33563  bnj1384  33568
  Copyright terms: Public domain W3C validator