Theorem fvpr2g

Description: The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017) (Proof shortened by BJ, 26-Sep-2024)

	Ref	Expression
Assertion	fvpr2g	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵 ) → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		prcom	⊢ { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } = { ⟨ 𝐵 , 𝐷 ⟩ , ⟨ 𝐴 , 𝐶 ⟩ }
2	1	fveq1i	⊢ ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐵 ) = ( { ⟨ 𝐵 , 𝐷 ⟩ , ⟨ 𝐴 , 𝐶 ⟩ } ‘ 𝐵 )
3		necom	⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴 )
4		fvpr1g	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐵 ≠ 𝐴 ) → ( { ⟨ 𝐵 , 𝐷 ⟩ , ⟨ 𝐴 , 𝐶 ⟩ } ‘ 𝐵 ) = 𝐷 )
5	3 4	syl3an3b	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵 ) → ( { ⟨ 𝐵 , 𝐷 ⟩ , ⟨ 𝐴 , 𝐶 ⟩ } ‘ 𝐵 ) = 𝐷 )
6	2 5	eqtrid	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵 ) → ( { ⟨ 𝐴 , 𝐶 ⟩ , ⟨ 𝐵 , 𝐷 ⟩ } ‘ 𝐵 ) = 𝐷 )