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Metamath Proof Explorer

Theorem elrelsrel

Description: The element of the relations class ( df-rels ) and the relation predicate are the same when R is a set. (Contributed by Peter Mazsa, 24-Nov-2018)

		Ref	Expression
	Assertion	elrelsrel	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		elrels2	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ 𝑅 ⊆ ( V × V ) ) )
2		df-rel	⊢ ( Rel 𝑅 ↔ 𝑅 ⊆ ( V × V ) )
3	1 2	bitr4di	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )