Theorem usgrfun

Description: The edge function of a simple graph is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017) (Revised by AV, 13-Oct-2020)

	Ref	Expression
Assertion	usgrfun	$$⊢G\in \mathrm{USGraph}\to \mathrm{Fun}\mathrm{iEdg}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(G\right)$$
2		eqid	$$⊢\mathrm{iEdg}\left(G\right)=\mathrm{iEdg}\left(G\right)$$
3	1 2	usgrfs	$$⊢G\in \mathrm{USGraph}\to \mathrm{iEdg}\left(G\right):\mathrm{dom}\mathrm{iEdg}\left(G\right)\underset{1-1}{⟶}\left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$
4		f1fun	$$⊢\mathrm{iEdg}\left(G\right):\mathrm{dom}\mathrm{iEdg}\left(G\right)\underset{1-1}{⟶}\left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}\to \mathrm{Fun}\mathrm{iEdg}\left(G\right)$$
5	3 4	syl	$$⊢G\in \mathrm{USGraph}\to \mathrm{Fun}\mathrm{iEdg}\left(G\right)$$