Theorem fvtp3

Description: The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	fvtp3.1	$$⊢C\in \mathrm{V}$$
	fvtp3.4	$$⊢F\in \mathrm{V}$$
Assertion	fvtp3	$$⊢\left(A\ne C\wedge B\ne C\right)\to \left\{⟨A,D⟩,⟨B,E⟩,⟨C,F⟩\right\}\left(C\right)=F$$

Proof

Step	Hyp	Ref	Expression
1		fvtp3.1	$$⊢C\in \mathrm{V}$$
2		fvtp3.4	$$⊢F\in \mathrm{V}$$
3		tprot	$$⊢\left\{⟨A,D⟩,⟨B,E⟩,⟨C,F⟩\right\}=\left\{⟨B,E⟩,⟨C,F⟩,⟨A,D⟩\right\}$$
4	3	fveq1i	$$⊢\left\{⟨A,D⟩,⟨B,E⟩,⟨C,F⟩\right\}\left(C\right)=\left\{⟨B,E⟩,⟨C,F⟩,⟨A,D⟩\right\}\left(C\right)$$
5		necom	$$⊢A\ne C\leftrightarrow C\ne A$$
6	1 2	fvtp2	$$⊢\left(B\ne C\wedge C\ne A\right)\to \left\{⟨B,E⟩,⟨C,F⟩,⟨A,D⟩\right\}\left(C\right)=F$$
7	5 6	sylan2b	$$⊢\left(B\ne C\wedge A\ne C\right)\to \left\{⟨B,E⟩,⟨C,F⟩,⟨A,D⟩\right\}\left(C\right)=F$$
8	7	ancoms	$$⊢\left(A\ne C\wedge B\ne C\right)\to \left\{⟨B,E⟩,⟨C,F⟩,⟨A,D⟩\right\}\left(C\right)=F$$
9	4 8	syl5eq	$$⊢\left(A\ne C\wedge B\ne C\right)\to \left\{⟨A,D⟩,⟨B,E⟩,⟨C,F⟩\right\}\left(C\right)=F$$