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Theorem csbima12gALT 5206
Description: Move class substitution in and out of the image of a function. (This is csbima12g 5205 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbima12gALTVD 28946. Although the proof is shorter, the total number of steps of all theorems used in the proof is probably longer. (Contributed by NM, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbima12gALT  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )

Proof of Theorem csbima12gALT
StepHypRef Expression
1 csbrng 5106 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  [_ A  /  x ]_ ( F  |`  B ) )
2 csbresg 5141 . . . 4  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F  |`  B )  =  (
[_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
32rneqd 5089 . . 3  |-  ( A  e.  C  ->  ran  [_ A  /  x ]_ ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |` 
[_ A  /  x ]_ B ) )
41, 3eqtrd 2467 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ran  ( F  |`  B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B ) )
5 df-ima 4883 . . 3  |-  ( F
" B )  =  ran  ( F  |`  B )
65csbeq2i 3269 . 2  |-  [_ A  /  x ]_ ( F
" B )  = 
[_ A  /  x ]_ ran  ( F  |`  B )
7 df-ima 4883 . 2  |-  ( [_ A  /  x ]_ F "
[_ A  /  x ]_ B )  =  ran  ( [_ A  /  x ]_ F  |`  [_ A  /  x ]_ B )
84, 6, 73eqtr4g 2492 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 1652    e. wcel 1725   [_csb 3243   ran crn 4871    |` cres 4872   "cima 4873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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