Theorem cosselcnvrefrels3

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 30-Aug-2021)

	Ref	Expression
Assertion	cosselcnvrefrels3	|- ( ,~ R e. CnvRefRels <-> ( A. u A. x A. y ( ( u R x /\ u R y ) -> x = y ) /\ ,~ R e. Rels ) )

Proof

Step	Hyp	Ref	Expression
1		cosselcnvrefrels2	|- ( ,~ R e. CnvRefRels <-> ( ,~ R C_ _I /\ ,~ R e. Rels ) )
2		cossssid3	|- ( ,~ R C_ _I <-> A. u A. x A. y ( ( u R x /\ u R y ) -> x = y ) )
3	2	anbi1i	|- ( ( ,~ R C_ _I /\ ,~ R e. Rels ) <-> ( A. u A. x A. y ( ( u R x /\ u R y ) -> x = y ) /\ ,~ R e. Rels ) )
4	1 3	bitri	|- ( ,~ R e. CnvRefRels <-> ( A. u A. x A. y ( ( u R x /\ u R y ) -> x = y ) /\ ,~ R e. Rels ) )