Theorem cosselcnvrefrels4

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 31-Aug-2021)

	Ref	Expression
Assertion	cosselcnvrefrels4	|- ( ,~ R e. CnvRefRels <-> ( A. u E* x u R x /\ ,~ R e. Rels ) )

Proof

Step	Hyp	Ref	Expression
1		cosselcnvrefrels2	|- ( ,~ R e. CnvRefRels <-> ( ,~ R C_ _I /\ ,~ R e. Rels ) )
2		cossssid4	|- ( ,~ R C_ _I <-> A. u E* x u R x )
3	2	anbi1i	|- ( ( ,~ R C_ _I /\ ,~ R e. Rels ) <-> ( A. u E* x u R x /\ ,~ R e. Rels ) )
4	1 3	bitri	|- ( ,~ R e. CnvRefRels <-> ( A. u E* x u R x /\ ,~ R e. Rels ) )