Theorem 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)

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

Proof

Step	Hyp	Ref	Expression
1		dfrel2	⊢ ( Rel 𝑅 ↔ ◡ ◡ 𝑅 = 𝑅 )
2		eqcom	⊢ ( ◡ ◡ 𝑅 = 𝑅 ↔ 𝑅 = ◡ ◡ 𝑅 )
3		cnvcnv3	⊢ ◡ ◡ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 }
4	3	eqeq2i	⊢ ( 𝑅 = ◡ ◡ 𝑅 ↔ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 } )
5	1 2 4	3bitri	⊢ ( Rel 𝑅 ↔ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 } )