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	⊢ ( 𝜑 → 𝑆 ∈ 𝑊 )
	uhgrspan.q	⊢ ( 𝜑 → ( Vtx ‘ 𝑆 ) = 𝑉 )
	uhgrspan.r	⊢ ( 𝜑 → ( iEdg ‘ 𝑆 ) = ( 𝐸 ↾ 𝐴 ) )
	uhgrspan.g	⊢ ( 𝜑 → 𝐺 ∈ UHGraph )
Assertion	uhgrspan	⊢ ( 𝜑 → 𝑆 ∈ UHGraph )

Proof

Step	Hyp	Ref	Expression
1		uhgrspan.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		uhgrspan.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
3		uhgrspan.s	⊢ ( 𝜑 → 𝑆 ∈ 𝑊 )
4		uhgrspan.q	⊢ ( 𝜑 → ( Vtx ‘ 𝑆 ) = 𝑉 )
5		uhgrspan.r	⊢ ( 𝜑 → ( iEdg ‘ 𝑆 ) = ( 𝐸 ↾ 𝐴 ) )
6		uhgrspan.g	⊢ ( 𝜑 → 𝐺 ∈ UHGraph )
7	1 2 3 4 5 6	uhgrspansubgr	⊢ ( 𝜑 → 𝑆 SubGraph 𝐺 )
8		subuhgr	⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺 ) → 𝑆 ∈ UHGraph )
9	6 7 8	syl2anc	⊢ ( 𝜑 → 𝑆 ∈ UHGraph )