Theorem elrefrelsrel

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

	Ref	Expression
Assertion	elrefrelsrel	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ RefRels ↔ RefRel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )
2	1	anbi2d	⊢ ( 𝑅 ∈ 𝑉 → ( ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) ) )
3		elrefrels2	⊢ ( 𝑅 ∈ RefRels ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) )
4		dfrefrel2	⊢ ( RefRel 𝑅 ↔ ( ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ 𝑅 ∧ Rel 𝑅 ) )
5	2 3 4	3bitr4g	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ RefRels ↔ RefRel 𝑅 ) )