Theorem breldm

Description: Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995)

	Ref	Expression
Hypotheses	opeldm.1	$$⊢A\in \mathrm{V}$$
	opeldm.2	$$⊢B\in \mathrm{V}$$
Assertion	breldm	$$⊢ARB\to A\in \mathrm{dom}R$$

Proof

Step	Hyp	Ref	Expression
1		opeldm.1	$$⊢A\in \mathrm{V}$$
2		opeldm.2	$$⊢B\in \mathrm{V}$$
3		df-br	$$⊢ARB\leftrightarrow ⟨A,B⟩\in R$$
4	1 2	opeldm	$$⊢⟨A,B⟩\in R\to A\in \mathrm{dom}R$$
5	3 4	sylbi	$$⊢ARB\to A\in \mathrm{dom}R$$