Theorem csbdm 5318

Description: Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)

Assertion

Ref	Expression
csbdm	|- [_ A / x ]_ dom B = dom [_ A / x ]_ B

Proof of Theorem csbdm

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

Step	Hyp	Ref	Expression
1		csbab 4008	. . 3 |- [_ A / x ]_ { y | E. w <. y , w >. e. B } = { y | [. A / x ]. E. w <. y , w >. e. B }
2		sbcex2 3486	. . . . 5 |- ( [. A / x ]. E. w <. y , w >. e. B <-> E. w [. A / x ]. <. y , w >. e. B )
3		sbcel2 3989	. . . . . 6 |- ( [. A / x ]. <. y , w >. e. B <-> <. y , w >. e. [_ A / x ]_ B )
4	3	exbii 1774	. . . . 5 |- ( E. w [. A / x ]. <. y , w >. e. B <-> E. w <. y , w >. e. [_ A / x ]_ B )
5	2, 4	bitri 264	. . . 4 |- ( [. A / x ]. E. w <. y , w >. e. B <-> E. w <. y , w >. e. [_ A / x ]_ B )
6	5	abbii 2739	. . 3 |- { y | [. A / x ]. E. w <. y , w >. e. B } = { y | E. w <. y , w >. e. [_ A / x ]_ B }
7	1, 6	eqtri 2644	. 2 |- [_ A / x ]_ { y | E. w <. y , w >. e. B } = { y | E. w <. y , w >. e. [_ A / x ]_ B }
8		dfdm3 5310	. . 3 |- dom B = { y | E. w <. y , w >. e. B }
9	8	csbeq2i 3993	. 2 |- [_ A / x ]_ dom B = [_ A / x ]_ { y | E. w <. y , w >. e. B }
10		dfdm3 5310	. 2 |- dom [_ A / x ]_ B = { y | E. w <. y , w >. e. [_ A / x ]_ B }
11	7, 9, 10	3eqtr4i 2654	1 |- [_ A / x ]_ dom B = dom [_ A / x ]_ B

Syntax hints:   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533   <.cop 4183   dom cdm 5114

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916  df-br 4654  df-dm 5124

This theorem is referenced by:  sbcfng  6042