Theorem csbfv12gALTOLD 33329

Description: Move class substitution in and out of a function value. The proof is derived from the virtual deduction proof csbfv12gALTVD 33407. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 5887 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 4259	. . 3 |- ( A e. C -> [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. [_ A / x ]_ { y | ( F " { B } ) = { y } } )
2		csbabgOLD 3838	. . . . 5 |- ( A e. C -> [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | [. A / x ]. ( F " { B } ) = { y } } )
3		sbceqg 3807	. . . . . . 7 |- ( A e. C -> ( [. A / x ]. ( F " { B } ) = { y } <-> [_ A / x ]_ ( F " { B } ) = [_ A / x ]_ { y } ) )
4		csbima12gOLD 5341	. . . . . . . . 9 |- ( A e. C -> [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " [_ A / x ]_ { B } ) )
5		csbsng 4069	. . . . . . . . . 10 |- ( A e. C -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
6	5	imaeq2d 5323	. . . . . . . . 9 |- ( A e. C -> ( [_ A / x ]_ F " [_ A / x ]_ { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) )
7	4, 6	eqtrd 2482	. . . . . . . 8 |- ( A e. C -> [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) )
8		csbconstg 3430	. . . . . . . 8 |- ( A e. C -> [_ A / x ]_ { y } = { y } )
9	7, 8	eqeq12d 2463	. . . . . . 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 2577	. . . . 5 |- ( A e. C -> { y | [. A / x ]. ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
12	2, 11	eqtrd 2482	. . . 4 |- ( A e. C -> [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
13	12	unieqd 4240	. . 3 |- ( A e. C -> U. [_ A / x ]_ { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
14	1, 13	eqtrd 2482	. 2 |- ( A e. C -> [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
15		dffv4 5849	. . 3 |- ( F ` B ) = U. { y | ( F " { B } ) = { y } }
16	15	csbeq2i 3818	. 2 |- [_ A / x ]_ ( F ` B ) = [_ A / x ]_ U. { y | ( F " { B } ) = { y } }
17		dffv4 5849	. 2 |- ( [_ A / x ]_ F ` [_ A / x ]_ B ) = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } }
18	14, 16, 17	3eqtr4g 2507	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1381   e. wcel 1802   {cab 2426   [.wsbc 3311   [_csb 3417   {csn 4010   U.cuni 4230   "cima 4988   ` cfv 5574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-xp 4991  df-cnv 4993  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fv 5582

This theorem is referenced by: (None)