Theorem upgrres1lem2

Description: Lemma 2 for upgrres1 . (Contributed by AV, 7-Nov-2020)

	Ref	Expression
Hypotheses	upgrres1.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	upgrres1.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
	upgrres1.f	$$⊢F=\left\{e\in E|N\notin e\right\}$$
	upgrres1.s	$$⊢S=⟨V\setminus \left\{N\right\},{\mathrm{I}\upharpoonright }_F⟩$$
Assertion	upgrres1lem2	$$⊢\mathrm{Vtx}\left(S\right)=V\setminus \left\{N\right\}$$

Proof

Step	Hyp	Ref	Expression
1		upgrres1.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		upgrres1.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
3		upgrres1.f	$$⊢F=\left\{e\in E|N\notin e\right\}$$
4		upgrres1.s	$$⊢S=⟨V\setminus \left\{N\right\},{\mathrm{I}\upharpoonright }_F⟩$$
5	4	fveq2i	$$⊢\mathrm{Vtx}\left(S\right)=\mathrm{Vtx}\left(⟨V\setminus \left\{N\right\},{\mathrm{I}\upharpoonright }_F⟩\right)$$
6	1 2 3	upgrres1lem1	$$⊢\left(V\setminus \left\{N\right\}\in \mathrm{V}\wedge {\mathrm{I}\upharpoonright }_F\in \mathrm{V}\right)$$
7		opvtxfv	$$⊢\left(V\setminus \left\{N\right\}\in \mathrm{V}\wedge {\mathrm{I}\upharpoonright }_F\in \mathrm{V}\right)\to \mathrm{Vtx}\left(⟨V\setminus \left\{N\right\},{\mathrm{I}\upharpoonright }_F⟩\right)=V\setminus \left\{N\right\}$$
8	6 7	ax-mp	$$⊢\mathrm{Vtx}\left(⟨V\setminus \left\{N\right\},{\mathrm{I}\upharpoonright }_F⟩\right)=V\setminus \left\{N\right\}$$
9	5 8	eqtri	$$⊢\mathrm{Vtx}\left(S\right)=V\setminus \left\{N\right\}$$