Theorem brovpreldm 7254

Description: If a binary relation holds for the result of an operation, the operands are in the domain of the operation. (Contributed by AV, 31-Dec-2020.)

Assertion

Ref	Expression
brovpreldm	|- ( D ( B A C ) E -> <. B , C >. e. dom A )

Proof of Theorem brovpreldm

Step	Hyp	Ref	Expression
1		df-br 4654	. 2 |- ( D ( B A C ) E <-> <. D , E >. e. ( B A C ) )
2		ne0i 3921	. . 3 |- ( <. D , E >. e. ( B A C ) -> ( B A C ) =/= (/) )
3		df-ov 6653	. . . . 5 |- ( B A C ) = ( A ` <. B , C >. )
4		ndmfv 6218	. . . . 5 |- ( -. <. B , C >. e. dom A -> ( A ` <. B , C >. ) = (/) )
5	3, 4	syl5eq 2668	. . . 4 |- ( -. <. B , C >. e. dom A -> ( B A C ) = (/) )
6	5	necon1ai 2821	. . 3 |- ( ( B A C ) =/= (/) -> <. B , C >. e. dom A )
7	2, 6	syl 17	. 2 |- ( <. D , E >. e. ( B A C ) -> <. B , C >. e. dom A )
8	1, 7	sylbi 207	1 |- ( D ( B A C ) E -> <. B , C >. e. dom A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  bropopvvv  7255  bropfvvvv  7257