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Metamath Proof Explorer

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	$⊢ V = Vtx (G)$
	uhgrf.e	$⊢ E = iEdg (G)$
Assertion	ushgrf	$⊢ G \in USHGraph \to E : dom E \underset{1-1}{⟶} 𝒫 V ∖ \{\emptyset\}$

Proof

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