Theorem eqvrelcl

Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015) (Revised by Peter Mazsa, 2-Jun-2019)

	Ref	Expression
Hypotheses	eqvrelcl.1	|- ( ph -> EqvRel R )
	eqvrelcl.2	|- ( ph -> A R B )
Assertion	eqvrelcl	|- ( ph -> A e. dom R )

Proof

Step	Hyp	Ref	Expression
1		eqvrelcl.1	|- ( ph -> EqvRel R )
2		eqvrelcl.2	|- ( ph -> A R B )
3		eqvrelrel	|- ( EqvRel R -> Rel R )
4	1 3	syl	|- ( ph -> Rel R )
5		releldm	|- ( ( Rel R /\ A R B ) -> A e. dom R )
6	4 2 5	syl2anc	|- ( ph -> A e. dom R )