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	⊢ ( 𝐴 ≠ 𝐵 → ( { ⟨ 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 ∧ 𝐴 ≠ 𝐵 ) → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐷 )
4	1 2 3	mp3an12	⊢ ( 𝐴 ≠ 𝐵 → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐷 )