Theorem csbfv12gALTOLD 39052

Description: Move class substitution in and out of a function value. The proof is derived from the virtual deduction proof csbfv12gALTVD 39135. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 6231 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		csbuni 4466	. . . 4 |- [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. [_ A / x ]_ { y | ( F " { B } ) = { y } }
2	1	a1i 11	. . 3 |- ( A e. C -> [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. [_ A / x ]_ { y | ( F " { B } ) = { y } } )
3		csbab 4008	. . . . . 6 |- [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | [. A / x ]. ( F " { B } ) = { y } }
4	3	a1i 11	. . . . 5 |- ( A e. C -> [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | [. A / x ]. ( F " { B } ) = { y } } )
5		sbceqg 3984	. . . . . . 7 |- ( A e. C -> ( [. A / x ]. ( F " { B } ) = { y } <-> [_ A / x ]_ ( F " { B } ) = [_ A / x ]_ { y } ) )
6		csbima12 5483	. . . . . . . . . 10 |- [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " [_ A / x ]_ { B } )
7	6	a1i 11	. . . . . . . . 9 |- ( A e. C -> [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " [_ A / x ]_ { B } ) )
8		csbsng 4243	. . . . . . . . . 10 |- ( A e. C -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
9	8	imaeq2d 5466	. . . . . . . . 9 |- ( A e. C -> ( [_ A / x ]_ F " [_ A / x ]_ { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) )
10	7, 9	eqtrd 2656	. . . . . . . 8 |- ( A e. C -> [_ A / x ]_ ( F " { B } ) = ( [_ A / x ]_ F " { [_ A / x ]_ B } ) )
11		csbconstg 3546	. . . . . . . 8 |- ( A e. C -> [_ A / x ]_ { y } = { y } )
12	10, 11	eqeq12d 2637	. . . . . . 7 |- ( A e. C -> ( [_ A / x ]_ ( F " { B } ) = [_ A / x ]_ { y } <-> ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } ) )
13	5, 12	bitrd 268	. . . . . 6 |- ( A e. C -> ( [. A / x ]. ( F " { B } ) = { y } <-> ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } ) )
14	13	abbidv 2741	. . . . 5 |- ( A e. C -> { y | [. A / x ]. ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
15	4, 14	eqtrd 2656	. . . 4 |- ( A e. C -> [_ A / x ]_ { y | ( F " { B } ) = { y } } = { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
16	15	unieqd 4446	. . 3 |- ( A e. C -> U. [_ A / x ]_ { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
17	2, 16	eqtrd 2656	. 2 |- ( A e. C -> [_ A / x ]_ U. { y | ( F " { B } ) = { y } } = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } } )
18		dffv4 6188	. . 3 |- ( F ` B ) = U. { y | ( F " { B } ) = { y } }
19	18	csbeq2i 3993	. 2 |- [_ A / x ]_ ( F ` B ) = [_ A / x ]_ U. { y | ( F " { B } ) = { y } }
20		dffv4 6188	. 2 |- ( [_ A / x ]_ F ` [_ A / x ]_ B ) = U. { y | ( [_ A / x ]_ F " { [_ A / x ]_ B } ) = { y } }
21	17, 19, 20	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   {cab 2608   [.wsbc 3435   [_csb 3533   {csn 4177   U.cuni 4436   "cima 5117   ` cfv 5888

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-8 1992  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-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  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-ral 2917  df-rex 2918  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-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896

This theorem is referenced by: (None)