Theorem csbrn 5304

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 5191	. . 3 |- [_ A / x ]_ ( B " _V ) = ( [_ A / x ]_ B " [_ A / x ]_ _V )
2		csbconstg 3362	. . . . 5 |- ( A e. _V -> [_ A / x ]_ _V = _V )
3	2	imaeq2d 5174	. . . 4 |- ( A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( [_ A / x ]_ B " _V ) )
4		0ima 5190	. . . . . 6 |- ( (/) " _V ) = (/)
5	4	eqcomi 2480	. . . . 5 |- (/) = ( (/) " _V )
6		csbprc 3774	. . . . . . 7 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
7	6	imaeq1d 5173	. . . . . 6 |- ( -. A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( (/) " [_ A / x ]_ _V ) )
8		0ima 5190	. . . . . 6 |- ( (/) " [_ A / x ]_ _V ) = (/)
9	7, 8	syl6eq 2521	. . . . 5 |- ( -. A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = (/) )
10	6	imaeq1d 5173	. . . . 5 |- ( -. A e. _V -> ( [_ A / x ]_ B " _V ) = ( (/) " _V ) )
11	5, 9, 10	3eqtr4a 2531	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( [_ A / x ]_ B " _V ) )
12	3, 11	pm2.61i 169	. . 3 |- ( [_ A / x ]_ B " [_ A / x ]_ _V ) = ( [_ A / x ]_ B " _V )
13	1, 12	eqtri 2493	. 2 |- [_ A / x ]_ ( B " _V ) = ( [_ A / x ]_ B " _V )
14		dfrn4 5303	. . 3 |- ran B = ( B " _V )
15	14	csbeq2i 3786	. 2 |- [_ A / x ]_ ran B = [_ A / x ]_ ( B " _V )
16		dfrn4 5303	. 2 |- ran [_ A / x ]_ B = ( [_ A / x ]_ B " _V )
17	13, 15, 16	3eqtr4i 2503	1 |- [_ A / x ]_ ran B = ran [_ A / x ]_ B

Syntax hints:   -. wn 3   = wceq 1452   e. wcel 1904   _Vcvv 3031   [_csb 3349   (/)c0 3722   ran crn 4840   "cima 4842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852

This theorem is referenced by:  sbcfg  5737  nbgraopALT  25231