Theorem eqvreldmqs2

Description: Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 30-Dec-2024)

	Ref	Expression
Assertion	eqvreldmqs2	|- ( ( EqvRel ,~ ( `' _E |` A ) /\ ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A ) <-> ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		df-coels	|- ~ A = ,~ ( `' _E |` A )
2	1	eqvreleqi	|- ( EqvRel ~ A <-> EqvRel ,~ ( `' _E |` A ) )
3	2	bicomi	|- ( EqvRel ,~ ( `' _E |` A ) <-> EqvRel ~ A )
4		dmqs1cosscnvepreseq	|- ( ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A <-> ( U. A /. ~ A ) = A )
5	3 4	anbi12i	|- ( ( EqvRel ,~ ( `' _E |` A ) /\ ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A ) <-> ( EqvRel ~ A /\ ( U. A /. ~ A ) = A ) )