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	$$⊢B\in \mathrm{V}$$
	fvpr2.2	$$⊢D\in \mathrm{V}$$
Assertion	fvpr2	$$⊢A\ne B\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=D$$

Proof

Step	Hyp	Ref	Expression
1		fvpr2.1	$$⊢B\in \mathrm{V}$$
2		fvpr2.2	$$⊢D\in \mathrm{V}$$
3		fvpr2g	$$⊢\left(B\in \mathrm{V}\wedge D\in \mathrm{V}\wedge A\ne B\right)\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=D$$
4	1 2 3	mp3an12	$$⊢A\ne B\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=D$$