Theorem csbima12g 5097

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 3130	. . 3 |- ( y = A -> [_ y / x ]_ ( F " B ) = [_ A / x ]_ ( F " B ) )
2		csbeq1 3130	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3130	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4	2, 3	imaeq12d 5077	. . 3 |- ( y = A -> ( [_ y / x ]_ F " [_ y / x ]_ B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
5	1, 4	eqeq12d 2246	. 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 2805	. . 3 |- y e. _V
7		nfcsb1v 3160	. . . 4 |- F/_ x [_ y / x ]_ F
8		nfcsb1v 3160	. . . 4 |- F/_ x [_ y / x ]_ B
9	7, 8	nfima 5084	. . 3 |- F/_ x ( [_ y / x ]_ F " [_ y / x ]_ B )
10		csbeq1a 3136	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
11		csbeq1a 3136	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
12	10, 11	imaeq12d 5077	. . 3 |- ( x = y -> ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) )
13	6, 9, 12	csbief 3172	. 2 |- [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B )
14	5, 13	vtoclg 2864	1 |- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   [_csb 3127   "cima 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by: (None)