Metamath Proof Explorer


Theorem fv1prop

Description: The function value of unordered pair of ordered pairs with first components 1 and 2 at 1. (Contributed by AV, 4-Feb-2023)

Ref Expression
Assertion fv1prop ( 𝐴𝑉 → ( { ⟨ 1 , 𝐴 ⟩ , ⟨ 2 , 𝐵 ⟩ } ‘ 1 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 1ex 1 ∈ V
2 1ne2 1 ≠ 2
3 fvpr1g ( ( 1 ∈ V ∧ 𝐴𝑉 ∧ 1 ≠ 2 ) → ( { ⟨ 1 , 𝐴 ⟩ , ⟨ 2 , 𝐵 ⟩ } ‘ 1 ) = 𝐴 )
4 1 2 3 mp3an13 ( 𝐴𝑉 → ( { ⟨ 1 , 𝐴 ⟩ , ⟨ 2 , 𝐵 ⟩ } ‘ 1 ) = 𝐴 )