Theorem elcoeleqvrels

Description: Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023)

	Ref	Expression
Assertion	elcoeleqvrels	|- ( A e. V -> ( A e. CoElEqvRels <-> ,~ ( `' _E |` A ) e. EqvRels ) )

Proof

Step	Hyp	Ref	Expression
1		reseq2	|- ( a = A -> ( `' _E |` a ) = ( `' _E |` A ) )
2	1	cosseqd	|- ( a = A -> ,~ ( `' _E |` a ) = ,~ ( `' _E |` A ) )
3	2	eleq1d	|- ( a = A -> ( ,~ ( `' _E |` a ) e. EqvRels <-> ,~ ( `' _E |` A ) e. EqvRels ) )
4		df-coeleqvrels	|- CoElEqvRels = { a | ,~ ( `' _E |` a ) e. EqvRels }
5	3 4	elab2g	|- ( A e. V -> ( A e. CoElEqvRels <-> ,~ ( `' _E |` A ) e. EqvRels ) )