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Theorem csbima12gALTOLD 32702
Description: Move class substitution in and out of the image of a function. The proof is derived from the virtual deduction proof csbima12gALTVD 32777. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 5899 instead. (Proof modification is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbima12gALTOLD  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )

Proof of Theorem csbima12gALTOLD
StepHypRef Expression
1 csbrngOLD 5467 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) )
2 csbresgOLD 5275 . . . 4  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F  |`  B )  =  (
[_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
32rneqd 5228 . . 3  |-  ( A  e.  C  ->  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |` 
[_ A  /  x ]_ B ) )
41, 3eqtrd 2508 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
5 df-ima 5012 . . 3  |-  ( F
" B )  =  ran  ( F  |`  B )
65csbeq2i 3836 . 2  |-  [_ A  /  x ]_ ( F
" B )  = 
[_ A  /  x ]_ ran  ( F  |`  B )
7 df-ima 5012 . 2  |-  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )
84, 6, 73eqtr4g 2533 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   [_csb 3435   ran crn 5000    |` cres 5001   "cima 5002
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  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-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by: (None)
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