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	$$⊢E=\mathrm{iEdg}\left(G\right)$$
Assertion	usgrf1	$$⊢G\in \mathrm{USGraph}\to E:\mathrm{dom}E\underset{1-1}{⟶}\mathrm{ran}E$$

Proof

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