Theorem umgrfn

Description: The edge function of an undirected multigraph is a function into unordered pairs of vertices. (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	umgrfn	$$⊢\left(G\in \mathrm{UMGraph}\wedge EFnA\right)\to E:A⟶\left\{x\in 𝒫V|\left|x\right|=2\right\}$$

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	umgrf	$$⊢G\in \mathrm{UMGraph}\to E:\mathrm{dom}E⟶\left\{x\in 𝒫V|\left|x\right|=2\right\}$$
4		fndm	$$⊢EFnA\to \mathrm{dom}E=A$$
5	4	feq2d	$$⊢EFnA\to \left(E:\mathrm{dom}E⟶\left\{x\in 𝒫V|\left|x\right|=2\right\}\leftrightarrow E:A⟶\left\{x\in 𝒫V|\left|x\right|=2\right\}\right)$$
6	3 5	syl5ibcom	$$⊢G\in \mathrm{UMGraph}\to \left(EFnA\to E:A⟶\left\{x\in 𝒫V|\left|x\right|=2\right\}\right)$$
7	6	imp	$$⊢\left(G\in \mathrm{UMGraph}\wedge EFnA\right)\to E:A⟶\left\{x\in 𝒫V|\left|x\right|=2\right\}$$