Theorem elcnvrefrelsrel

Description: For sets, being an element of the class of converse reflexive relations ( df-cnvrefrels ) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021)

	Ref	Expression
Assertion	elcnvrefrelsrel	|- ( R e. V -> ( R e. CnvRefRels <-> CnvRefRel R ) )

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	|- ( R e. V -> ( R e. Rels <-> Rel R ) )
2	1	anbi2d	|- ( R e. V -> ( ( R C_ ( _I i^i ( dom R X. ran R ) ) /\ R e. Rels ) <-> ( R C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R ) ) )
3		elcnvrefrels2	|- ( R e. CnvRefRels <-> ( R C_ ( _I i^i ( dom R X. ran R ) ) /\ R e. Rels ) )
4		dfcnvrefrel2	|- ( CnvRefRel R <-> ( R C_ ( _I i^i ( dom R X. ran R ) ) /\ Rel R ) )
5	2 3 4	3bitr4g	|- ( R e. V -> ( R e. CnvRefRels <-> CnvRefRel R ) )