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) Avoid ax-10 , ax-12 . (Revised by SN, 7-Jun-2025)

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

Proof

Step	Hyp	Ref	Expression
1		relcnv	⊢ Rel ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
2		relopabv	⊢ Rel { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 }
3		elopab	⊢ ( ⟨ 𝑤 , 𝑧 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑤 , 𝑧 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
4		excom	⊢ ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑤 , 𝑧 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑦 ∃ 𝑥 ( ⟨ 𝑤 , 𝑧 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
5		ancom	⊢ ( ( 𝑤 = 𝑥 ∧ 𝑧 = 𝑦 ) ↔ ( 𝑧 = 𝑦 ∧ 𝑤 = 𝑥 ) )
6		vex	⊢ 𝑤 ∈ V
7		vex	⊢ 𝑧 ∈ V
8	6 7	opth	⊢ ( ⟨ 𝑤 , 𝑧 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ( 𝑤 = 𝑥 ∧ 𝑧 = 𝑦 ) )
9	7 6	opth	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑦 , 𝑥 ⟩ ↔ ( 𝑧 = 𝑦 ∧ 𝑤 = 𝑥 ) )
10	5 8 9	3bitr4i	⊢ ( ⟨ 𝑤 , 𝑧 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑦 , 𝑥 ⟩ )
11	10	anbi1i	⊢ ( ( ⟨ 𝑤 , 𝑧 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑦 , 𝑥 ⟩ ∧ 𝜑 ) )
12	11	2exbii	⊢ ( ∃ 𝑦 ∃ 𝑥 ( ⟨ 𝑤 , 𝑧 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑦 ∃ 𝑥 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑦 , 𝑥 ⟩ ∧ 𝜑 ) )
13	3 4 12	3bitri	⊢ ( ⟨ 𝑤 , 𝑧 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑦 ∃ 𝑥 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑦 , 𝑥 ⟩ ∧ 𝜑 ) )
14	7 6	opelcnv	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ⟨ 𝑤 , 𝑧 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } )
15		elopab	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 } ↔ ∃ 𝑦 ∃ 𝑥 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑦 , 𝑥 ⟩ ∧ 𝜑 ) )
16	13 14 15	3bitr4i	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 } )
17	1 2 16	eqrelriiv	⊢ ◡ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 }