Theorem csbrn 5452

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 5342	. . 3 |- [_ A / x ]_ ( B " _V ) = ( [_ A / x ]_ B " [_ A / x ]_ _V )
2		csbconstg 3433	. . . . 5 |- ( A e. _V -> [_ A / x ]_ _V = _V )
3	2	imaeq2d 5325	. . . 4 |- ( A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( [_ A / x ]_ B " _V ) )
4		0ima 5341	. . . . . 6 |- ( (/) " _V ) = (/)
5	4	eqcomi 2467	. . . . 5 |- (/) = ( (/) " _V )
6		csbprc 3820	. . . . . . 7 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
7	6	imaeq1d 5324	. . . . . 6 |- ( -. A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( (/) " [_ A / x ]_ _V ) )
8		0ima 5341	. . . . . 6 |- ( (/) " [_ A / x ]_ _V ) = (/)
9	7, 8	syl6eq 2511	. . . . 5 |- ( -. A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = (/) )
10	6	imaeq1d 5324	. . . . 5 |- ( -. A e. _V -> ( [_ A / x ]_ B " _V ) = ( (/) " _V ) )
11	5, 9, 10	3eqtr4a 2521	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( [_ A / x ]_ B " _V ) )
12	3, 11	pm2.61i 164	. . 3 |- ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( [_ A / x ]_ B " _V )
13	1, 12	eqtri 2483	. 2 |- [_ A / x ]_ ( B " _V ) = ( [_ A / x ]_ B " _V )
14		dfrn4 5451	. . 3 |- ran B = ( B " _V )
15	14	csbeq2i 3832	. 2 |- [_ A / x ]_ ran B = [_ A / x ]_ ( B " _V )
16		dfrn4 5451	. 2 |- ran [_ A / x ]_ B = ( [_ A / x ]_ B " _V )
17	13, 15, 16	3eqtr4i 2493	1 |- [_ A / x ]_ ran B = ran [_ A / x ]_ B

Syntax hints:   -. wn 3   = wceq 1398   e. wcel 1823   _Vcvv 3106   [_csb 3420   (/)c0 3783   ran crn 4989   "cima 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001

This theorem is referenced by:  sbcfg  5711  nbgraopALT  24626