Theorem csbima12 5483

Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.)

Assertion

Ref	Expression
csbima12	|- [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B )

Proof of Theorem csbima12

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . . 4 |- ( y = A -> [_ y / x ]_ ( F " B ) = [_ A / x ]_ ( F " B ) )
2		csbeq1 3536	. . . . 5 |- ( y = A -> [_ y / x ]_ F = [_ A / x ]_ F )
3		csbeq1 3536	. . . . 5 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
4	2, 3	imaeq12d 5467	. . . 4 |- ( y = A -> ( [_ y / x ]_ F " [_ y / x ]_ B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
5	1, 4	eqeq12d 2637	. . 3 |- ( y = A -> ( [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) <-> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) ) )
6		vex 3203	. . . 4 |- y e. _V
7		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ F
8		nfcsb1v 3549	. . . . 5 |- F/_ x [_ y / x ]_ B
9	7, 8	nfima 5474	. . . 4 |- F/_ x ( [_ y / x ]_ F " [_ y / x ]_ B )
10		csbeq1a 3542	. . . . 5 |- ( x = y -> F = [_ y / x ]_ F )
11		csbeq1a 3542	. . . . 5 |- ( x = y -> B = [_ y / x ]_ B )
12	10, 11	imaeq12d 5467	. . . 4 |- ( x = y -> ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B ) )
13	6, 9, 12	csbief 3558	. . 3 |- [_ y / x ]_ ( F " B ) = ( [_ y / x ]_ F " [_ y / x ]_ B )
14	5, 13	vtoclg 3266	. 2 |- ( A e. _V -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
15		csbprc 3980	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( F " B ) = (/) )
16		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
17	16	imaeq2d 5466	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ F " [_ A / x ]_ B ) = ( [_ A / x ]_ F " (/) ) )
18		ima0 5481	. . . 4 |- ( [_ A / x ]_ F " (/) ) = (/)
19	17, 18	syl6req 2673	. . 3 |- ( -. A e. _V -> (/) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
20	15, 19	eqtrd 2656	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
21	14, 20	pm2.61i 176	1 |- [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   (/)c0 3915   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  csbrn  5596  disjpreima  29397  csbpredg  33172  brtrclfv2  38019  sbcheg  38073  csbfv12gALTOLD  39052  csbfv12gALTVD  39135