Theorem eqvreldmqs

Description: Two ways to express membership equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021) (Revised by Peter Mazsa, 17-Jul-2023)

	Ref	Expression
Assertion	eqvreldmqs	|- ( ( EqvRel ,~ ( `' _E |` A ) /\ ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A ) <-> ( CoElEqvRel A /\ ( U. A /. ~ A ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		df-coeleqvrel	|- ( CoElEqvRel A <-> EqvRel ,~ ( `' _E |` A ) )
2	1	bicomi	|- ( EqvRel ,~ ( `' _E |` A ) <-> CoElEqvRel A )
3		dmqs1cosscnvepreseq	|- ( ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A <-> ( U. A /. ~ A ) = A )
4	2 3	anbi12i	|- ( ( EqvRel ,~ ( `' _E |` A ) /\ ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A ) <-> ( CoElEqvRel A /\ ( U. A /. ~ A ) = A ) )