Theorem csbima12g 5143

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 3150	. . 3 |- ( y = A -> [_ y / x ]_ ( F " B ) = [_ A / x ]_ ( F " B ) )
2		csbeq1 3150	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3150	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4	2, 3	imaeq12d 5122	. . 3 |- ( y = A -> ( [_ y / x ]_ F " [_ y / x ]_ B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
5	1, 4	eqeq12d 2253	. 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 2824	. . 3 |- y e. _V
7		nfcsb1v 3180	. . . 4 |- F/_ x [_ y / x ]_ F
8		nfcsb1v 3180	. . . 4 |- F/_ x [_ y / x ]_ B
9	7, 8	nfima 5129	. . 3 |- F/_ x ( [_ y / x ]_ F " [_ y / x ]_ B )
10		csbeq1a 3156	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
11		csbeq1a 3156	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
12	10, 11	imaeq12d 5122	. . 3 |- ( x = y -> ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) )
13	6, 9, 12	csbief 3192	. 2 |- [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B )
14	5, 13	vtoclg 2883	1 |- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   [_csb 3147   "cima 4772

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by: (None)