Theorem csbfv12gALTOLD 31414

Description: Move class substitution in and out of a function value. The proof is derived from the virtual deduction proof csbfv12gALTVD 31492. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 5718 instead. (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbfv12gALTOLD	|- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Proof of Theorem csbfv12gALTOLD

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbunigOLD 4113	. . 3 |- ( A e. C -> [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. [_ A / x ]_ { y | ( F " { B } ) = { y } } )
2		csbabgOLD 3701	. . . . 5 |- ( A e. C -> [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | [. A / x ]. ( F " { B } ) = { y } } )
3		sbceqg 3670	. . . . . . 7 |- ( A e. C -> ( [. A / x ]. ( F " { B } ) = { y } <-> [_ A / x ]_ ( F " { B } ) = [_ A / x ]_ { y } ) )
4		csbima12gOLD 5180	. . . . . . . . 9 |- ( A e. C -> [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " [_ A / x ]_ { B } ) )
5		csbsng 3928	. . . . . . . . . 10 |- ( A e. C -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
6	5	imaeq2d 5162	. . . . . . . . 9 |- ( A e. C -> ( [_ A / x ]_ F " [_ A / x ]_ { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) )
7	4, 6	eqtrd 2469	. . . . . . . 8 |- ( A e. C -> [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) )
8		csbconstg 3294	. . . . . . . 8 |- ( A e. C -> [_ A / x ]_ { y } = { y } )
9	7, 8	eqeq12d 2451	. . . . . . 7 |- ( A e. C -> ( [_ A / x ]_ ( F " { B } ) = [_ A / x ]_ { y } <-> ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } ) )
10	3, 9	bitrd 253	. . . . . 6 |- ( A e. C -> ( [. A / x ]. ( F " { B } ) = { y } <-> ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } ) )
11	10	abbidv 2551	. . . . 5 |- ( A e. C -> { y | [. A / x ]. ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
12	2, 11	eqtrd 2469	. . . 4 |- ( A e. C -> [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
13	12	unieqd 4094	. . 3 |- ( A e. C -> U. [_ A / x ]_ { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
14	1, 13	eqtrd 2469	. 2 |- ( A e. C -> [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
15		dffv4 5681	. . 3 |- ( F ` B ) = U. { y | ( F " { B } ) = { y } }
16	15	csbeq2i 3681	. 2 |- [_ A / x ]_ ( F ` B ) = [_ A / x ]_ U. { y | ( F " { B } ) = { y } }
17		dffv4 5681	. 2 |- ( [_ A / x ]_ F ` [_ A / x ]_ B ) = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } }
18	14, 16, 17	3eqtr4g 2494	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1369   e. wcel 1756   {cab 2423   [.wsbc 3179   [_csb 3281   {csn 3870   U.cuni 4084   "cima 4835   ` cfv 5411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-br 4286  df-opab 4344  df-xp 4838  df-cnv 4840  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fv 5419

This theorem is referenced by: (None)