Theorem csbima12g 4965

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 3048	. . 3 |- ( y = A -> [_ y / x ]_ ( F " B ) = [_ A / x ]_ ( F " B ) )
2		csbeq1 3048	. . . 4 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3048	. . . 4 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4	2, 3	imaeq12d 4947	. . 3 |- ( y = A -> ( [_ y / x ]_ F " [_ y / x ]_ B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
5	1, 4	eqeq12d 2180	. 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 2729	. . 3 |- y e. _V
7		nfcsb1v 3078	. . . 4 |- F/_ x [_ y / x ]_ F
8		nfcsb1v 3078	. . . 4 |- F/_ x [_ y / x ]_ B
9	7, 8	nfima 4954	. . 3 |- F/_ x ( [_ y / x ]_ F " [_ y / x ]_ B )
10		csbeq1a 3054	. . . 4 |- ( x = y -> F = [_ y / x ]_ F )
11		csbeq1a 3054	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
12	10, 11	imaeq12d 4947	. . 3 |- ( x = y -> ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) )
13	6, 9, 12	csbief 3089	. 2 |- [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B )
14	5, 13	vtoclg 2786	1 |- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   [_csb 3045   "cima 4607

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by: (None)