Theorem dfmember2

Description: Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021)

	Ref	Expression
Assertion	dfmember2	$$⊢MembErA\leftrightarrow \left(EqvRel\sim A\wedge \mathrm{dom}\sim A/\sim A=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfmember	$$⊢MembErA\leftrightarrow \sim AErALTVA$$
2		dferALTV2	$$⊢\sim AErALTVA\leftrightarrow \left(EqvRel\sim A\wedge \mathrm{dom}\sim A/\sim A=A\right)$$
3	1 2	bitri	$$⊢MembErA\leftrightarrow \left(EqvRel\sim A\wedge \mathrm{dom}\sim A/\sim A=A\right)$$