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Metamath Proof Explorer

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 }