Theorem csbima12g 5062

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 3104	. . 3 |- ( y = A -> [_ y / x ]_ ( F " B ) = [_ A / x ]_ ( F " B ) )
2		csbeq1 3104	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3104	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4	2, 3	imaeq12d 5042	. . 3 |- ( y = A -> ( [_ y / x ]_ F " [_ y / x ]_ B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
5	1, 4	eqeq12d 2222	. 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 2779	. . 3 |- y e. _V
7		nfcsb1v 3134	. . . 4 |- F/_ x [_ y / x ]_ F
8		nfcsb1v 3134	. . . 4 |- F/_ x [_ y / x ]_ B
9	7, 8	nfima 5049	. . 3 |- F/_ x ( [_ y / x ]_ F " [_ y / x ]_ B )
10		csbeq1a 3110	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
11		csbeq1a 3110	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
12	10, 11	imaeq12d 5042	. . 3 |- ( x = y -> ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) )
13	6, 9, 12	csbief 3146	. 2 |- [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B )
14	5, 13	vtoclg 2838	1 |- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   [_csb 3101   "cima 4696

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by: (None)