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)

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 vopelopabsb ( ⟨ 𝑤 , 𝑧 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ [ 𝑤 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 )
4 sbcom2 ( [ 𝑤 / 𝑥 ] [ 𝑧 / 𝑦 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] [ 𝑤 / 𝑥 ] 𝜑 )
5 3 4 bitri ( ⟨ 𝑤 , 𝑧 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ [ 𝑧 / 𝑦 ] [ 𝑤 / 𝑥 ] 𝜑 )
6 vex 𝑧 ∈ V
7 vex 𝑤 ∈ V
8 6 7 opelcnv ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ⟨ 𝑤 , 𝑧 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } )
9 vopelopabsb ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑦 , 𝑥 ⟩ ∣ 𝜑 } ↔ [ 𝑧 / 𝑦 ] [ 𝑤 / 𝑥 ] 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 }