Theorem dmqs1cosscnvepreseq

Description: Two ways to express the equality of the domain quotient of the coelements on the class A with the class A . (Contributed by Peter Mazsa, 26-Sep-2021)

	Ref	Expression
Assertion	dmqs1cosscnvepreseq	|- ( ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A <-> ( U. A /. ~ A ) = A )

Proof

Step	Hyp	Ref	Expression
1		df-coels	|- ~ A = ,~ ( `' _E |` A )
2	1	dmqseqeq1i	|- ( ( dom ~ A /. ~ A ) = A <-> ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A )
3		dmqscoelseq	|- ( ( dom ~ A /. ~ A ) = A <-> ( U. A /. ~ A ) = A )
4	2 3	bitr3i	|- ( ( dom ,~ ( `' _E |` A ) /. ,~ ( `' _E |` A ) ) = A <-> ( U. A /. ~ A ) = A )