Theorem upgrres1lem1

Description: Lemma 1 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	$$⊢F=\left\{e\in E|N\notin e\right\}$$
Assertion	upgrres1lem1	$$⊢\left(V\setminus \left\{N\right\}\in \mathrm{V}\wedge {\mathrm{I}\upharpoonright }_F\in \mathrm{V}\right)$$

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	1	fvexi	$$⊢V\in \mathrm{V}$$
5	4	difexi	$$⊢V\setminus \left\{N\right\}\in \mathrm{V}$$
6	2	fvexi	$$⊢E\in \mathrm{V}$$
7	3 6	rabex2	$$⊢F\in \mathrm{V}$$
8		resiexg	$$⊢F\in \mathrm{V}\to {\mathrm{I}\upharpoonright }_F\in \mathrm{V}$$
9	7 8	ax-mp	$$⊢{\mathrm{I}\upharpoonright }_F\in \mathrm{V}$$
10	5 9	pm3.2i	$$⊢\left(V\setminus \left\{N\right\}\in \mathrm{V}\wedge {\mathrm{I}\upharpoonright }_F\in \mathrm{V}\right)$$