Theorem csbfv12gALTOLD 32576

Description: Move class substitution in and out of a function value. The proof is derived from the virtual deduction proof csbfv12gALTVD 32654. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 5892 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 4267	. . 3 |- ( A e. C -> [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. [_ A / x ]_ { y | ( F " { B } ) = { y } } )
2		csbabgOLD 3849	. . . . 5 |- ( A e. C -> [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | [. A / x ]. ( F " { B } ) = { y } } )
3		sbceqg 3818	. . . . . . 7 |- ( A e. C -> ( [. A / x ]. ( F " { B } ) = { y } <-> [_ A / x ]_ ( F " { B } ) = [_ A / x ]_ { y } ) )
4		csbima12gOLD 5346	. . . . . . . . 9 |- ( A e. C -> [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " [_ A / x ]_ { B } ) )
5		csbsng 4079	. . . . . . . . . 10 |- ( A e. C -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
6	5	imaeq2d 5328	. . . . . . . . 9 |- ( A e. C -> ( [_ A / x ]_ F " [_ A / x ]_ { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) )
7	4, 6	eqtrd 2501	. . . . . . . 8 |- ( A e. C -> [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) )
8		csbconstg 3441	. . . . . . . 8 |- ( A e. C -> [_ A / x ]_ { y } = { y } )
9	7, 8	eqeq12d 2482	. . . . . . 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 2596	. . . . 5 |- ( A e. C -> { y | [. A / x ]. ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
12	2, 11	eqtrd 2501	. . . 4 |- ( A e. C -> [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
13	12	unieqd 4248	. . 3 |- ( A e. C -> U. [_ A / x ]_ { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
14	1, 13	eqtrd 2501	. 2 |- ( A e. C -> [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
15		dffv4 5854	. . 3 |- ( F ` B ) = U. { y | ( F " { B } ) = { y } }
16	15	csbeq2i 3829	. 2 |- [_ A / x ]_ ( F ` B ) = [_ A / x ]_ U. { y | ( F " { B } ) = { y } }
17		dffv4 5854	. 2 |- ( [_ A / x ]_ F ` [_ A / x ]_ B ) = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } }
18	14, 16, 17	3eqtr4g 2526	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1374   e. wcel 1762   {cab 2445   [.wsbc 3324   [_csb 3428   {csn 4020   U.cuni 4238   "cima 4995   ` cfv 5579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fv 5587

This theorem is referenced by: (None)