Theorem uhgrspan1lem2

Description: Lemma 2 for uhgrspan1 . (Contributed by AV, 19-Nov-2020)

	Ref	Expression
Hypotheses	uhgrspan1.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	uhgrspan1.i	$$⊢I=\mathrm{iEdg}\left(G\right)$$
	uhgrspan1.f	$$⊢F=\left\{i\in \mathrm{dom}I|N\notin I\left(i\right)\right\}$$
	uhgrspan1.s	$$⊢S=⟨V\setminus \left\{N\right\},{I\upharpoonright }_F⟩$$
Assertion	uhgrspan1lem2	$$⊢\mathrm{Vtx}\left(S\right)=V\setminus \left\{N\right\}$$

Proof

Step	Hyp	Ref	Expression
1		uhgrspan1.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		uhgrspan1.i	$$⊢I=\mathrm{iEdg}\left(G\right)$$
3		uhgrspan1.f	$$⊢F=\left\{i\in \mathrm{dom}I|N\notin I\left(i\right)\right\}$$
4		uhgrspan1.s	$$⊢S=⟨V\setminus \left\{N\right\},{I\upharpoonright }_F⟩$$
5	4	fveq2i	$$⊢\mathrm{Vtx}\left(S\right)=\mathrm{Vtx}\left(⟨V\setminus \left\{N\right\},{I\upharpoonright }_F⟩\right)$$
6	1 2 3	uhgrspan1lem1	$$⊢\left(V\setminus \left\{N\right\}\in \mathrm{V}\wedge {I\upharpoonright }_F\in \mathrm{V}\right)$$
7		opvtxfv	$$⊢\left(V\setminus \left\{N\right\}\in \mathrm{V}\wedge {I\upharpoonright }_F\in \mathrm{V}\right)\to \mathrm{Vtx}\left(⟨V\setminus \left\{N\right\},{I\upharpoonright }_F⟩\right)=V\setminus \left\{N\right\}$$
8	6 7	ax-mp	$$⊢\mathrm{Vtx}\left(⟨V\setminus \left\{N\right\},{I\upharpoonright }_F⟩\right)=V\setminus \left\{N\right\}$$
9	5 8	eqtri	$$⊢\mathrm{Vtx}\left(S\right)=V\setminus \left\{N\right\}$$