Theorem uhgrspan1lem1

Description: Lemma 1 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	$$⊢F=\left\{i\in \mathrm{dom}I|N\notin I\left(i\right)\right\}$$
Assertion	uhgrspan1lem1	$$⊢\left(V\setminus \left\{N\right\}\in \mathrm{V}\wedge {I\upharpoonright }_F\in \mathrm{V}\right)$$

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	1	fvexi	$$⊢V\in \mathrm{V}$$
5	4	difexi	$$⊢V\setminus \left\{N\right\}\in \mathrm{V}$$
6	2	fvexi	$$⊢I\in \mathrm{V}$$
7	6	resex	$$⊢{I\upharpoonright }_F\in \mathrm{V}$$
8	5 7	pm3.2i	$$⊢\left(V\setminus \left\{N\right\}\in \mathrm{V}\wedge {I\upharpoonright }_F\in \mathrm{V}\right)$$