Theorem umgrf

Description: The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfn without explicitly specified domain of the edge function. (Contributed by AV, 24-Nov-2020)

	Ref	Expression
Hypotheses	isumgr.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	isumgr.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
Assertion	umgrf	$$⊢G\in \mathrm{UMGraph}\to E:\mathrm{dom}E⟶\left\{x\in 𝒫V|\left|x\right|=2\right\}$$

Proof

Step	Hyp	Ref	Expression
1		isumgr.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		isumgr.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
3	1 2	isumgrs	$$⊢G\in \mathrm{UMGraph}\to \left(G\in \mathrm{UMGraph}\leftrightarrow E:\mathrm{dom}E⟶\left\{x\in 𝒫V|\left|x\right|=2\right\}\right)$$
4	3	ibi	$$⊢G\in \mathrm{UMGraph}\to E:\mathrm{dom}E⟶\left\{x\in 𝒫V|\left|x\right|=2\right\}$$