Theorem usgrspan

Description: A spanning subgraph S of a simple graph G is a simple graph. (Contributed by AV, 15-Oct-2020) (Revised by AV, 16-Oct-2020) (Proof shortened by AV, 18-Nov-2020)

	Ref	Expression
Hypotheses	uhgrspan.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	uhgrspan.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
	uhgrspan.s	$$⊢\phi \to S\in W$$
	uhgrspan.q	$$⊢\phi \to \mathrm{Vtx}\left(S\right)=V$$
	uhgrspan.r	$$⊢\phi \to \mathrm{iEdg}\left(S\right)={E\upharpoonright }_A$$
	usgrspan.g	$$⊢\phi \to G\in \mathrm{USGraph}$$
Assertion	usgrspan	$$⊢\phi \to S\in \mathrm{USGraph}$$

Proof

Step	Hyp	Ref	Expression
1		uhgrspan.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		uhgrspan.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
3		uhgrspan.s	$$⊢\phi \to S\in W$$
4		uhgrspan.q	$$⊢\phi \to \mathrm{Vtx}\left(S\right)=V$$
5		uhgrspan.r	$$⊢\phi \to \mathrm{iEdg}\left(S\right)={E\upharpoonright }_A$$
6		usgrspan.g	$$⊢\phi \to G\in \mathrm{USGraph}$$
7		usgruhgr	$$⊢G\in \mathrm{USGraph}\to G\in \mathrm{UHGraph}$$
8	6 7	syl	$$⊢\phi \to G\in \mathrm{UHGraph}$$
9	1 2 3 4 5 8	uhgrspansubgr	$$⊢\phi \to S\mathrm{SubGraph}G$$
10		subusgr	$$⊢\left(G\in \mathrm{USGraph}\wedge S\mathrm{SubGraph}G\right)\to S\in \mathrm{USGraph}$$
11	6 9 10	syl2anc	$$⊢\phi \to S\in \mathrm{USGraph}$$