Theorem ushgrf

Description: The edge function of an undirected simple hypergraph is a one-to-one function into the power set of the set of vertices. (Contributed by AV, 9-Oct-2020)

	Ref	Expression
Hypotheses	uhgrf.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	uhgrf.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
Assertion	ushgrf	$$⊢G\in \mathrm{USHGraph}\to E:\mathrm{dom}E\underset{1-1}{⟶}𝒫V\setminus \left\{\varnothing \right\}$$

Proof

Step	Hyp	Ref	Expression
1		uhgrf.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		uhgrf.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
3	1 2	isushgr	$$⊢G\in \mathrm{USHGraph}\to \left(G\in \mathrm{USHGraph}\leftrightarrow E:\mathrm{dom}E\underset{1-1}{⟶}𝒫V\setminus \left\{\varnothing \right\}\right)$$
4	3	ibi	$$⊢G\in \mathrm{USHGraph}\to E:\mathrm{dom}E\underset{1-1}{⟶}𝒫V\setminus \left\{\varnothing \right\}$$