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Theorem csbima12g 5022
Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
csbima12g  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )

Proof of Theorem csbima12g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3084 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( F
" B )  = 
[_ A  /  x ]_ ( F " B
) )
2 csbeq1 3084 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3084 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
42, 3imaeq12d 5013 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
51, 4eqeq12d 2297 . 2  |-  ( y  =  A  ->  ( [_ y  /  x ]_ ( F " B
)  =  ( [_ y  /  x ]_ F "
[_ y  /  x ]_ B )  <->  [_ A  /  x ]_ ( F " B )  =  (
[_ A  /  x ]_ F " [_ A  /  x ]_ B ) ) )
6 vex 2791 . . 3  |-  y  e. 
_V
7 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ F
8 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ B
97, 8nfima 5020 . . 3  |-  F/_ x
( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
10 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
11 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
1210, 11imaeq12d 5013 . . 3  |-  ( x  =  y  ->  ( F " B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B ) )
136, 9, 12csbief 3122 . 2  |-  [_ y  /  x ]_ ( F
" B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
145, 13vtoclg 2843 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   [_csb 3081   "cima 4692
This theorem is referenced by:  csbfv12gALT  5536  csbfv12gALTVD  28048
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  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-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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