Theorem cosselcnvrefrels5

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 5-Sep-2021)

	Ref	Expression
Assertion	cosselcnvrefrels5	|- ( ,~ R e. CnvRefRels <-> ( A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) /\ ,~ R e. Rels ) )

Proof

Step	Hyp	Ref	Expression
1		cosselcnvrefrels2	|- ( ,~ R e. CnvRefRels <-> ( ,~ R C_ _I /\ ,~ R e. Rels ) )
2		cossssid5	|- ( ,~ R C_ _I <-> A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) )
3	2	anbi1i	|- ( ( ,~ R C_ _I /\ ,~ R e. Rels ) <-> ( A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) /\ ,~ R e. Rels ) )
4	1 3	bitri	|- ( ,~ R e. CnvRefRels <-> ( A. x e. ran R A. y e. ran R ( x = y \/ ( [ x ] `' R i^i [ y ] `' R ) = (/) ) /\ ,~ R e. Rels ) )