Theorem elrels2

Description: The element of the relations class ( df-rels ) and the relation predicate ( df-rel ) are the same when R is a set. (Contributed by Peter Mazsa, 14-Jun-2018)

	Ref	Expression
Assertion	elrels2	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ 𝑅 ⊆ ( V × V ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-rels	⊢ Rels = 𝒫 ( V × V )
2	1	eleq2i	⊢ ( 𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 ( V × V ) )
3		elpwg	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ 𝒫 ( V × V ) ↔ 𝑅 ⊆ ( V × V ) ) )
4	2 3	syl5bb	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ 𝑅 ⊆ ( V × V ) ) )