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 ∈ 𝑉 → ( 𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )
2	1	anbi2d	⊢ ( 𝑅 ∈ 𝑉 → ( ( 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ 𝑅 ∈ Rels ) ↔ ( 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) ) )
3		elcnvrefrels2	⊢ ( 𝑅 ∈ CnvRefRels ↔ ( 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ 𝑅 ∈ Rels ) )
4		dfcnvrefrel2	⊢ ( CnvRefRel 𝑅 ↔ ( 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) )
5	2 3 4	3bitr4g	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅 ) )