Theorem csbima12g 4908

Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
csbima12g	|- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )

Proof of Theorem csbima12g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3010	. . 3 |- ( y = A -> [_ y / x ]_ ( F " B ) = [_ A / x ]_ ( F " B ) )
2		csbeq1 3010	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3010	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4	2, 3	imaeq12d 4890	. . 3 |- ( y = A -> ( [_ y / x ]_ F " [_ y / x ]_ B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
5	1, 4	eqeq12d 2155	. 2 |- ( y = A -> ( [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) <-> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) ) )
6		vex 2692	. . 3 |- y e. _V
7		nfcsb1v 3040	. . . 4 |- F/_ x [_ y / x ]_ F
8		nfcsb1v 3040	. . . 4 |- F/_ x [_ y / x ]_ B
9	7, 8	nfima 4897	. . 3 |- F/_ x ( [_ y / x ]_ F " [_ y / x ]_ B )
10		csbeq1a 3016	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
11		csbeq1a 3016	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
12	10, 11	imaeq12d 4890	. . 3 |- ( x = y -> ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) )
13	6, 9, 12	csbief 3049	. 2 |- [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B )
14	5, 13	vtoclg 2749	1 |- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   [_csb 3007   "cima 4550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by: (None)