Theorem brovpreldm

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)

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

Proof

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