Theorem upgrf

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Hypotheses	isupgr.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	isupgr.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
Assertion	upgrf	$$⊢G\in \mathrm{UPGraph}\to E:\mathrm{dom}E⟶\left\{x\in \left(𝒫V\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$

Proof

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