Theorem bj-dfrel4v

Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 for relations. (Contributed by Mario Carneiro, 16-Aug-2015) Do not depend on df-cnv , reverse equality. (Revised by BJ, 16-Oct-2026) Note: this theorem could be used in place of opabid2 , which could be deleted.

	Ref	Expression
Assertion	bj-dfrel4v	⊢ ( Rel 𝑅 ↔ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 } = 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		bj-dfrel3	⊢ ( Rel 𝑅 ↔ ( 𝑅 ↾ V ) = 𝑅 )
2		relpart	⊢ ( 𝑅 ↾ V ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 }
3	2	eqeq1i	⊢ ( ( 𝑅 ↾ V ) = 𝑅 ↔ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 } = 𝑅 )
4	1 3	bitri	⊢ ( Rel 𝑅 ↔ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 } = 𝑅 )