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)

	Ref	Expression
Assertion	fvpr2g	$$⊢\left(B\in V\wedge D\in W\wedge A\ne B\right)\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=D$$

Proof

Step	Hyp	Ref	Expression
1		prcom	$$⊢\left\{⟨A,C⟩,⟨B,D⟩\right\}=\left\{⟨B,D⟩,⟨A,C⟩\right\}$$
2		df-pr	$$⊢\left\{⟨B,D⟩,⟨A,C⟩\right\}=\left\{⟨B,D⟩\right\}\cup \left\{⟨A,C⟩\right\}$$
3	1 2	eqtri	$$⊢\left\{⟨A,C⟩,⟨B,D⟩\right\}=\left\{⟨B,D⟩\right\}\cup \left\{⟨A,C⟩\right\}$$
4	3	fveq1i	$$⊢\left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=\left(\left\{⟨B,D⟩\right\}\cup \left\{⟨A,C⟩\right\}\right)\left(B\right)$$
5		fvunsn	$$⊢A\ne B\to \left(\left\{⟨B,D⟩\right\}\cup \left\{⟨A,C⟩\right\}\right)\left(B\right)=\left\{⟨B,D⟩\right\}\left(B\right)$$
6	4 5	syl5eq	$$⊢A\ne B\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=\left\{⟨B,D⟩\right\}\left(B\right)$$
7	6	3ad2ant3	$$⊢\left(B\in V\wedge D\in W\wedge A\ne B\right)\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=\left\{⟨B,D⟩\right\}\left(B\right)$$
8		fvsng	$$⊢\left(B\in V\wedge D\in W\right)\to \left\{⟨B,D⟩\right\}\left(B\right)=D$$
9	8	3adant3	$$⊢\left(B\in V\wedge D\in W\wedge A\ne B\right)\to \left\{⟨B,D⟩\right\}\left(B\right)=D$$
10	7 9	eqtrd	$$⊢\left(B\in V\wedge D\in W\wedge A\ne B\right)\to \left\{⟨A,C⟩,⟨B,D⟩\right\}\left(B\right)=D$$