Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015) (Proof shortened by BJ, 12-Feb-2022)
Ref | Expression | ||
---|---|---|---|
Assertion | cnvsng | ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ◡ { ⟨ 𝐴 , 𝐵 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnvsn | ⊢ ◡ ◡ { ⟨ 𝐵 , 𝐴 ⟩ } = ◡ { ⟨ 𝐴 , 𝐵 ⟩ } | |
2 | relsnopg | ⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ) → Rel { ⟨ 𝐵 , 𝐴 ⟩ } ) | |
3 | 2 | ancoms | ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → Rel { ⟨ 𝐵 , 𝐴 ⟩ } ) |
4 | dfrel2 | ⊢ ( Rel { ⟨ 𝐵 , 𝐴 ⟩ } ↔ ◡ ◡ { ⟨ 𝐵 , 𝐴 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } ) | |
5 | 3 4 | sylib | ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ◡ ◡ { ⟨ 𝐵 , 𝐴 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } ) |
6 | 1 5 | eqtr3id | ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ◡ { ⟨ 𝐴 , 𝐵 ⟩ } = { ⟨ 𝐵 , 𝐴 ⟩ } ) |