Theorem uhgrspan

Description: A spanning subgraph S of a hypergraph G is a hypergraph. (Contributed by AV, 11-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	$$⊢\phi \to \mathrm{iEdg}\left(S\right)={E\upharpoonright }_A$$
	uhgrspan.g	$$⊢\phi \to G\in \mathrm{UHGraph}$$
Assertion	uhgrspan	$$⊢\phi \to S\in \mathrm{UHGraph}$$

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		uhgrspan.g	$$⊢\phi \to G\in \mathrm{UHGraph}$$
7	1 2 3 4 5 6	uhgrspansubgr	$$⊢\phi \to S\mathrm{SubGraph}G$$
8		subuhgr	$$⊢\left(G\in \mathrm{UHGraph}\wedge S\mathrm{SubGraph}G\right)\to S\in \mathrm{UHGraph}$$
9	6 7 8	syl2anc	$$⊢\phi \to S\in \mathrm{UHGraph}$$