Theorem csbrngOLD 39056

Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5596 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbrngOLD	|- ( A e. V -> [_ A / x ]_ ran B = ran [_ A / x ]_ B )

Proof of Theorem csbrngOLD

Dummy variables w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbabgOLD 39050	. . 3 |- ( A e. V -> [_ A / x ]_ { y | E. w <. w , y >. e. B } = { y | [. A / x ]. E. w <. w , y >. e. B } )
2		sbcexgOLD 38753	. . . . 5 |- ( A e. V -> ( [. A / x ]. E. w <. w , y >. e. B <-> E. w [. A / x ]. <. w , y >. e. B ) )
3		sbcel2gOLD 38755	. . . . . 6 |- ( A e. V -> ( [. A / x ]. <. w , y >. e. B <-> <. w , y >. e. [_ A / x ]_ B ) )
4	3	exbidv 1850	. . . . 5 |- ( A e. V -> ( E. w [. A / x ]. <. w , y >. e. B <-> E. w <. w , y >. e. [_ A / x ]_ B ) )
5	2, 4	bitrd 268	. . . 4 |- ( A e. V -> ( [. A / x ]. E. w <. w , y >. e. B <-> E. w <. w , y >. e. [_ A / x ]_ B ) )
6	5	abbidv 2741	. . 3 |- ( A e. V -> { y | [. A / x ]. E. w <. w , y >. e. B } = { y | E. w <. w , y >. e. [_ A / x ]_ B } )
7	1, 6	eqtrd 2656	. 2 |- ( A e. V -> [_ A / x ]_ { y | E. w <. w , y >. e. B } = { y | E. w <. w , y >. e. [_ A / x ]_ B } )
8		dfrn3 5312	. . 3 |- ran B = { y | E. w <. w , y >. e. B }
9	8	csbeq2i 3993	. 2 |- [_ A / x ]_ ran B = [_ A / x ]_ { y | E. w <. w , y >. e. B }
10		dfrn3 5312	. 2 |- ran [_ A / x ]_ B = { y | E. w <. w , y >. e. [_ A / x ]_ B }
11	7, 9, 10	3eqtr4g 2681	1 |- ( A e. V -> [_ A / x ]_ ran B = ran [_ A / x ]_ B )

Syntax hints:   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533   <.cop 4183   ran crn 5115

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-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-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  csbima12gALTOLD  39057  csbima12gALTVD  39133