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Metamath Proof Explorer

Theorem usgrf1

Description: The edge function of a simple graph is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017) (Revised by AV, 15-Oct-2020)

		Ref	Expression
	Hypothesis	usgrf1o.e	$⊢ E = iEdg (G)$
	Assertion	usgrf1	$⊢ G \in USGraph \to E : dom E \underset{1-1}{⟶} ran E$

Proof

Step	Hyp	Ref	Expression
1		usgrf1o.e	$⊢ E = iEdg (G)$
2	1	usgrf1o	$⊢ G \in USGraph \to E : dom E \underset{1-1 onto}{⟶} ran E$
3		f1of1	$⊢ E : dom E \underset{1-1 onto}{⟶} ran E \to E : dom E \underset{1-1}{⟶} ran E$
4	2 3	syl	$⊢ G \in USGraph \to E : dom E \underset{1-1}{⟶} ran E$