Theorem cosselcnvrefrels2

Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 25-Aug-2021)

	Ref	Expression
Assertion	cosselcnvrefrels2	|- ( ,~ R e. CnvRefRels <-> ( ,~ R C_ _I /\ ,~ R e. Rels ) )

Proof

Step	Hyp	Ref	Expression
1		elcnvrefrels2	|- ( ,~ R e. CnvRefRels <-> ( ,~ R C_ ( _I i^i ( dom ,~ R X. ran ,~ R ) ) /\ ,~ R e. Rels ) )
2		cossssid	|- ( ,~ R C_ _I <-> ,~ R C_ ( _I i^i ( dom ,~ R X. ran ,~ R ) ) )
3	2	anbi1i	|- ( ( ,~ R C_ _I /\ ,~ R e. Rels ) <-> ( ,~ R C_ ( _I i^i ( dom ,~ R X. ran ,~ R ) ) /\ ,~ R e. Rels ) )
4	1 3	bitr4i	|- ( ,~ R e. CnvRefRels <-> ( ,~ R C_ _I /\ ,~ R e. Rels ) )