Theorem usgrf

Description: The edge function of a simple graph is a one-to-one function into the set of proper unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 13-Oct-2020)

	Ref	Expression
Hypotheses	isuspgr.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	isuspgr.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
Assertion	usgrf	$$⊢G\in \mathrm{USGraph}\to E:\mathrm{dom}E\underset{1-1}{⟶}\left\{x\in \left(𝒫V\setminus \left\{\varnothing \right\}\right)|\left|x\right|=2\right\}$$

Proof

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