Theorem fvpr2

Description: The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010)

	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		prcom	$$⊢\left\{⟨A,C⟩,⟨B,D⟩\right\}=\left\{⟨B,D⟩,⟨A,C⟩\right\}$$
4	3	fveq1i	$$⊢\left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=\left\{⟨B,D⟩,⟨A,C⟩\right\}\left(B\right)$$
5		necom	$$⊢A\ne B\leftrightarrow B\ne A$$
6	1 2	fvpr1	$$⊢B\ne A\to \left\{⟨B,D⟩,⟨A,C⟩\right\}\left(B\right)=D$$
7	5 6	sylbi	$$⊢A\ne B\to \left\{⟨B,D⟩,⟨A,C⟩\right\}\left(B\right)=D$$
8	4 7	eqtrid	$$⊢A\ne B\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=D$$