Theorem usgrf1o

Description: The edge function of a simple graph is a bijective function onto its range. (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	usgrf1o	$$⊢G\in \mathrm{USGraph}\to E:\mathrm{dom}E\underset{1-1\; onto}{⟶}\mathrm{ran}E$$

Proof

Step	Hyp	Ref	Expression
1		usgrf1o.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
2		eqid	$$⊢\mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(G\right)$$
3	2 1	usgrfs	$$⊢G\in \mathrm{USGraph}\to E:\mathrm{dom}E\underset{1-1}{⟶}\left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$
4		f1f1orn	$$⊢E:\mathrm{dom}E\underset{1-1}{⟶}\left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}\to E:\mathrm{dom}E\underset{1-1\; onto}{⟶}\mathrm{ran}E$$
5	3 4	syl	$$⊢G\in \mathrm{USGraph}\to E:\mathrm{dom}E\underset{1-1\; onto}{⟶}\mathrm{ran}E$$