Theorem csbima12gALTOLD 39057

Description: Move class substitution in and out of the image of a function. The proof is derived from the virtual deduction proof csbima12gALTVD 39133. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 6231 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

Step	Hyp	Ref	Expression
1		csbrngOLD 39056	. . 3 |- ( A e. C -> [_ A / x ]_ ran ( F |` B ) = ran [_ A / x ]_ ( F |` B ) )
2		csbresgOLD 39055	. . . 4 |- ( A e. C -> [_ A / x ]_ ( F |` B ) = ( [_ A / x ]_ F |` [_ A / x ]_ B ) )
3	2	rneqd 5353	. . 3 |- ( A e. C -> ran [_ A / x ]_ ( F |` B ) = ran ( [_ A / x ]_ F |` [_ A / x ]_ B ) )
4	1, 3	eqtrd 2656	. 2 |- ( A e. C -> [_ A / x ]_ ran ( F |` B ) = ran ( [_ A / x ]_ F |` [_ A / x ]_ B ) )
5		df-ima 5127	. . 3 |- ( F " B ) = ran ( F |` B )
6	5	csbeq2i 3993	. 2 |- [_ A / x ]_ ( F " B ) = [_ A / x ]_ ran ( F |` B )
7		df-ima 5127	. 2 |- ( [_ A / x ]_ F " [_ A / x ]_ B ) = ran ( [_ A / x ]_ F |` [_ A / x ]_ B )
8	4, 6, 7	3eqtr4g 2681	1 |- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   [_csb 3533   ran crn 5115   |` cres 5116   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by: (None)