Theorem csbrn 5596

Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

Ref	Expression
csbrn	|- [_ A / x ]_ ran B = ran [_ A / x ]_ B

Proof of Theorem csbrn

Step	Hyp	Ref	Expression
1		csbima12 5483	. . 3 |- [_ A / x ]_ ( B " _V ) = ( [_ A / x ]_ B " [_ A / x ]_ _V )
2		csbconstg 3546	. . . . 5 |- ( A e. _V -> [_ A / x ]_ _V = _V )
3	2	imaeq2d 5466	. . . 4 |- ( A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( [_ A / x ]_ B " _V ) )
4		0ima 5482	. . . . . 6 |- ( (/) " _V ) = (/)
5	4	eqcomi 2631	. . . . 5 |- (/) = ( (/) " _V )
6		csbprc 3980	. . . . . . 7 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
7	6	imaeq1d 5465	. . . . . 6 |- ( -. A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( (/) " [_ A / x ]_ _V ) )
8		0ima 5482	. . . . . 6 |- ( (/) " [_ A / x ]_ _V ) = (/)
9	7, 8	syl6eq 2672	. . . . 5 |- ( -. A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = (/) )
10	6	imaeq1d 5465	. . . . 5 |- ( -. A e. _V -> ( [_ A / x ]_ B " _V ) = ( (/) " _V ) )
11	5, 9, 10	3eqtr4a 2682	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( [_ A / x ]_ B " _V ) )
12	3, 11	pm2.61i 176	. . 3 |- ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( [_ A / x ]_ B " _V )
13	1, 12	eqtri 2644	. 2 |- [_ A / x ]_ ( B " _V ) = ( [_ A / x ]_ B " _V )
14		dfrn4 5595	. . 3 |- ran B = ( B " _V )
15	14	csbeq2i 3993	. 2 |- [_ A / x ]_ ran B = [_ A / x ]_ ( B " _V )
16		dfrn4 5595	. 2 |- ran [_ A / x ]_ B = ( [_ A / x ]_ B " _V )
17	13, 15, 16	3eqtr4i 2654	1 |- [_ A / x ]_ ran B = ran [_ A / x ]_ B

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   (/)c0 3915   ran crn 5115   "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-ral 2917  df-rex 2918  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-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  sbcfg  6043