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Theorem csbima12g 5003
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 3074 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( F
" B )  = 
[_ A  /  x ]_ ( F " B
) )
2 csbeq1 3074 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3074 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
42, 3imaeq12d 4985 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
51, 4eqeq12d 2203 . 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 2754 . . 3  |-  y  e. 
_V
7 nfcsb1v 3104 . . . 4  |-  F/_ x [_ y  /  x ]_ F
8 nfcsb1v 3104 . . . 4  |-  F/_ x [_ y  /  x ]_ B
97, 8nfima 4992 . . 3  |-  F/_ x
( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
10 csbeq1a 3080 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
11 csbeq1a 3080 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
1210, 11imaeq12d 4985 . . 3  |-  ( x  =  y  ->  ( F " B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B ) )
136, 9, 12csbief 3115 . 2  |-  [_ y  /  x ]_ ( F
" B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
145, 13vtoclg 2811 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 1363    e. wcel 2159   [_csb 3071   "cima 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-un 3147  df-in 3149  df-ss 3156  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-xp 4646  df-cnv 4648  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653
This theorem is referenced by: (None)
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