Theorem umgredgss

Description: The set of edges of a multigraph is a subset of the set of proper unordered pairs of vertices. (Contributed by AV, 25-Nov-2020)

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

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	umgrf	$$⊢G\in \mathrm{UMGraph}\to \mathrm{iEdg}\left(G\right):\mathrm{dom}\mathrm{iEdg}\left(G\right)⟶\left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$
5	4	frnd	$$⊢G\in \mathrm{UMGraph}\to \mathrm{ran}\mathrm{iEdg}\left(G\right)\subseteq \left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$
6	1 5	eqsstrid	$$⊢G\in \mathrm{UMGraph}\to \mathrm{Edg}\left(G\right)\subseteq \left\{x\in 𝒫\mathrm{Vtx}\left(G\right)|\left|x\right|=2\right\}$$