Theorem cnvcnv3

Description: The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015)

	Ref	Expression
Assertion	cnvcnv3	⊢ ◡ ◡ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 }

Proof

Step	Hyp	Ref	Expression
1		df-cnv	⊢ ◡ ◡ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 ◡ 𝑅 𝑥 }
2		vex	⊢ 𝑦 ∈ V
3		vex	⊢ 𝑥 ∈ V
4	2 3	brcnv	⊢ ( 𝑦 ◡ 𝑅 𝑥 ↔ 𝑥 𝑅 𝑦 )
5	4	opabbii	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 ◡ 𝑅 𝑥 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 }
6	1 5	eqtri	⊢ ◡ ◡ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 𝑅 𝑦 }