Theorem cnvopab

Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	cnvopab	⊢ ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		relcnv	⊢ Rel ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
2		relopab	⊢ Rel { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 }
3		opelopabsbALT	⊢ ( ⟨ 𝑤 , 𝑧 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ [ 𝑧 / 𝑦 ] [ 𝑤 / 𝑥 ] 𝜑 )
4		sbcom2	⊢ ( [ 𝑧 / 𝑦 ] [ 𝑤 / 𝑥 ] 𝜑 ↔ [ 𝑤 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 )
5	3 4	bitri	⊢ ( ⟨ 𝑤 , 𝑧 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ [ 𝑤 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 )
6		vex	⊢ 𝑧 ∈ V
7		vex	⊢ 𝑤 ∈ V
8	6 7	opelcnv	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ⟨ 𝑤 , 𝑧 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } )
9		opelopabsbALT	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 } ↔ [ 𝑤 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 )
10	5 8 9	3bitr4i	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 } )
11	1 2 10	eqrelriiv	⊢ ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 }