Metamath Proof Explorer

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	$⊢ D (B A C) E \to ⟨B, C⟩ \in dom A$

Proof

Step	Hyp	Ref	Expression
1		df-br	$⊢ D (B A C) E \leftrightarrow ⟨D, E⟩ \in (B A C)$
2		ne0i	$⊢ ⟨D, E⟩ \in (B A C) \to B A C \neq \emptyset$
3		df-ov	$⊢ B A C = A (⟨B, C⟩)$
4		ndmfv	$⊢ \neg ⟨B, C⟩ \in dom A \to A (⟨B, C⟩) = \emptyset$
5	3 4	eqtrid	$⊢ \neg ⟨B, C⟩ \in dom A \to B A C = \emptyset$
6	5	necon1ai	$⊢ B A C \neq \emptyset \to ⟨B, C⟩ \in dom A$
7	2 6	syl	$⊢ ⟨D, E⟩ \in (B A C) \to ⟨B, C⟩ \in dom A$
8	1 7	sylbi	$⊢ D (B A C) E \to ⟨B, C⟩ \in dom A$