Theorem usgredgss

Description: The set of edges of a simple graph is a subset of the set of proper unordered pairs of vertices. (Contributed by AV, 1-Jan-2020) (Revised by AV, 14-Oct-2020)

	Ref	Expression
Assertion	usgredgss	$$⊢G\in \mathrm{USGraph}\to \mathrm{Edg}\left(G\right)\subseteq \left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$

Proof

Step	Hyp	Ref	Expression
1		edgval	$$⊢\mathrm{Edg}\left(G\right)=\mathrm{ran}\mathrm{iEdg}\left(G\right)$$
2		eqid	$$⊢\mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(G\right)$$
3		eqid	$$⊢\mathrm{iEdg}\left(G\right)=\mathrm{iEdg}\left(G\right)$$
4	2 3	usgrfs	$$⊢G\in \mathrm{USGraph}\to \mathrm{iEdg}\left(G\right):\mathrm{dom}\mathrm{iEdg}\left(G\right)\underset{1-1}{⟶}\left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$
5		f1f	$$⊢\mathrm{iEdg}\left(G\right):\mathrm{dom}\mathrm{iEdg}\left(G\right)\underset{1-1}{⟶}\left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}\to \mathrm{iEdg}\left(G\right):\mathrm{dom}\mathrm{iEdg}\left(G\right)⟶\left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$
6		frn	$$⊢\mathrm{iEdg}\left(G\right):\mathrm{dom}\mathrm{iEdg}\left(G\right)⟶\left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}\to \mathrm{ran}\mathrm{iEdg}\left(G\right)\subseteq \left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$
7	4 5 6	3syl	$$⊢G\in \mathrm{USGraph}\to \mathrm{ran}\mathrm{iEdg}\left(G\right)\subseteq \left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$
8	1 7	eqsstrid	$$⊢G\in \mathrm{USGraph}\to \mathrm{Edg}\left(G\right)\subseteq \left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$