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Theorem acneq 7955
Description: Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acneq  |-  ( A  =  C  -> AC  A  = AC  C )

Proof of Theorem acneq
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2502 . . . 4  |-  ( A  =  C  ->  ( A  e.  _V  <->  C  e.  _V ) )
2 oveq2 6118 . . . . 5  |-  ( A  =  C  ->  (
( ~P x  \  { (/) } )  ^m  A )  =  ( ( ~P x  \  { (/) } )  ^m  C ) )
3 raleq 2910 . . . . . 6  |-  ( A  =  C  ->  ( A. y  e.  A  ( g `  y
)  e.  ( f `
 y )  <->  A. y  e.  C  ( g `  y )  e.  ( f `  y ) ) )
43exbidv 1637 . . . . 5  |-  ( A  =  C  ->  ( E. g A. y  e.  A  ( g `  y )  e.  ( f `  y )  <->  E. g A. y  e.  C  ( g `  y )  e.  ( f `  y ) ) )
52, 4raleqbidv 2922 . . . 4  |-  ( A  =  C  ->  ( A. f  e.  (
( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y )  <->  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  C ) E. g A. y  e.  C  ( g `  y
)  e.  ( f `
 y ) ) )
61, 5anbi12d 693 . . 3  |-  ( A  =  C  ->  (
( A  e.  _V  /\ 
A. f  e.  ( ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y ) )  <-> 
( C  e.  _V  /\ 
A. f  e.  ( ( ~P x  \  { (/) } )  ^m  C ) E. g A. y  e.  C  ( g `  y
)  e.  ( f `
 y ) ) ) )
76abbidv 2556 . 2  |-  ( A  =  C  ->  { x  |  ( A  e. 
_V  /\  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y ) ) }  =  { x  |  ( C  e. 
_V  /\  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  C ) E. g A. y  e.  C  ( g `  y
)  e.  ( f `
 y ) ) } )
8 df-acn 7860 . 2  |- AC  A  =  { x  |  ( A  e.  _V  /\  A. f  e.  ( ( ~P x  \  { (/)
} )  ^m  A
) E. g A. y  e.  A  (
g `  y )  e.  ( f `  y
) ) }
9 df-acn 7860 . 2  |- AC  C  =  { x  |  ( C  e.  _V  /\  A. f  e.  ( ( ~P x  \  { (/)
} )  ^m  C
) E. g A. y  e.  C  (
g `  y )  e.  ( f `  y
) ) }
107, 8, 93eqtr4g 2499 1  |-  ( A  =  C  -> AC  A  = AC  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1727   {cab 2428   A.wral 2711   _Vcvv 2962    \ cdif 3303   (/)c0 3613   ~Pcpw 3823   {csn 3838   ` cfv 5483  (class class class)co 6110    ^m cmap 7047  AC wacn 7856
This theorem is referenced by:  acndom  7963  dfacacn  8052  dfac13  8053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-iota 5447  df-fv 5491  df-ov 6113  df-acn 7860
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