Theorem bj-dfrel2

Description: Alternate definition of relation. Exercise 2 of TakeutiZaring p. 25. (Contributed by NM, 29-Dec-1996) (Proof shortened by BJ, 16-Oct-2026) Do not use opabid2 . (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-dfrel2	⊢ ( Rel 𝑅 ↔ ◡ ◡ 𝑅 = 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		bj-dfrel4v	⊢ ( Rel 𝑅 ↔ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 } = 𝑅 )
2		cnvcnv3	⊢ ◡ ◡ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 }
3	2	eqeq1i	⊢ ( ◡ ◡ 𝑅 = 𝑅 ↔ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 } = 𝑅 )
4	1 3	bitr4i	⊢ ( Rel 𝑅 ↔ ◡ ◡ 𝑅 = 𝑅 )