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Theorem csbima12g 4806
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 2937 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( F
" B )  = 
[_ A  /  x ]_ ( F " B
) )
2 csbeq1 2937 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 2937 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
42, 3imaeq12d 4788 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
51, 4eqeq12d 2103 . 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 2623 . . 3  |-  y  e. 
_V
7 nfcsb1v 2964 . . . 4  |-  F/_ x [_ y  /  x ]_ F
8 nfcsb1v 2964 . . . 4  |-  F/_ x [_ y  /  x ]_ B
97, 8nfima 4795 . . 3  |-  F/_ x
( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
10 csbeq1a 2942 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
11 csbeq1a 2942 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
1210, 11imaeq12d 4788 . . 3  |-  ( x  =  y  ->  ( F " B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B ) )
136, 9, 12csbief 2973 . 2  |-  [_ y  /  x ]_ ( F
" B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
145, 13vtoclg 2680 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 1290    e. wcel 1439   [_csb 2934   "cima 4455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465
This theorem is referenced by: (None)
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