Theorem fvpr1

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	fvpr1.1	$$⊢A\in \mathrm{V}$$
	fvpr1.2	$$⊢C\in \mathrm{V}$$
Assertion	fvpr1	$$⊢A\ne B\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(A\right)=C$$

Proof

Step	Hyp	Ref	Expression
1		fvpr1.1	$$⊢A\in \mathrm{V}$$
2		fvpr1.2	$$⊢C\in \mathrm{V}$$
3		fvpr1g	$$⊢\left(A\in \mathrm{V}\wedge C\in \mathrm{V}\wedge A\ne B\right)\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(A\right)=C$$
4	1 2 3	mp3an12	$$⊢A\ne B\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(A\right)=C$$