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Metamath Proof Explorer

Theorem usgrfs

Description: The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf . (Contributed by Alexander van der Vekens, 13-Aug-2017) (Revised by AV, 13-Oct-2020)

	Ref	Expression
Hypotheses	isuspgr.v	$⊢ V = Vtx (G)$
	isuspgr.e	$⊢ E = iEdg (G)$
Assertion	usgrfs	$⊢ G \in USGraph \to E : dom E \underset{1-1}{⟶} \{x \in 𝒫 V \| \|x\| = 2\}$

Proof

Step	Hyp	Ref	Expression
1		isuspgr.v	$⊢ V = Vtx (G)$
2		isuspgr.e	$⊢ E = iEdg (G)$
3	1 2	isusgrs	$⊢ G \in USGraph \to (G \in USGraph \leftrightarrow E : dom E \underset{1-1}{⟶} \{x \in 𝒫 V \| \|x\| = 2\})$
4	3	ibi	$⊢ G \in USGraph \to E : dom E \underset{1-1}{⟶} \{x \in 𝒫 V \| \|x\| = 2\}$