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Metamath Proof Explorer

Theorem usgredg2

Description: The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017) (Revised by AV, 16-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

		Ref	Expression
	Hypothesis	usgredg2.e	$⊢ E = iEdg (G)$
	Assertion	usgredg2	$⊢ (G \in USGraph \land X \in dom E) \to \|E (X)\| = 2$

Proof

Step	Hyp	Ref	Expression
1		usgredg2.e	$⊢ E = iEdg (G)$
2		usgrumgr	$⊢ G \in USGraph \to G \in UMGraph$
3		eqid	$⊢ Vtx (G) = Vtx (G)$
4	3 1	umgredg2	$⊢ (G \in UMGraph \land X \in dom E) \to \|E (X)\| = 2$
5	2 4	sylan	$⊢ (G \in USGraph \land X \in dom E) \to \|E (X)\| = 2$