Theorem elrels6

Description: Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021)

	Ref	Expression
Assertion	elrels6	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )
2		dfrel6	⊢ ( Rel 𝑅 ↔ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = 𝑅 )
3	1 2	bitrdi	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = 𝑅 ) )