Metamath Proof Explorer


Theorem fvpr2

Description: The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010) (Proof shortened by BJ, 26-Sep-2024)

Ref Expression
Hypotheses fvpr2.1 𝐵 ∈ V
fvpr2.2 𝐷 ∈ V
Assertion fvpr2 ( 𝐴𝐵 → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐷 )

Proof

Step Hyp Ref Expression
1 fvpr2.1 𝐵 ∈ V
2 fvpr2.2 𝐷 ∈ V
3 fvpr2g ( ( 𝐵 ∈ V ∧ 𝐷 ∈ V ∧ 𝐴𝐵 ) → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐷 )
4 1 2 3 mp3an12 ( 𝐴𝐵 → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐷 )