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 = ( Vtx ` G )
	uhgrspan.e	|- E = ( iEdg ` G )
	uhgrspan.s	|- ( ph -> S e. W )
	uhgrspan.q	|- ( ph -> ( Vtx ` S ) = V )
	uhgrspan.r	|- ( ph -> ( iEdg ` S ) = ( E |` A ) )
	uhgrspan.g	|- ( ph -> G e. UHGraph )
Assertion	uhgrspan	|- ( ph -> S e. UHGraph )

Proof

Step	Hyp	Ref	Expression
1		uhgrspan.v	|- V = ( Vtx ` G )
2		uhgrspan.e	|- E = ( iEdg ` G )
3		uhgrspan.s	|- ( ph -> S e. W )
4		uhgrspan.q	|- ( ph -> ( Vtx ` S ) = V )
5		uhgrspan.r	|- ( ph -> ( iEdg ` S ) = ( E |` A ) )
6		uhgrspan.g	|- ( ph -> G e. UHGraph )
7	1 2 3 4 5 6	uhgrspansubgr	|- ( ph -> S SubGraph G )
8		subuhgr	|- ( ( G e. UHGraph /\ S SubGraph G ) -> S e. UHGraph )
9	6 7 8	syl2anc	|- ( ph -> S e. UHGraph )