Theorem uhgrss

Description: An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017) (Revised by AV, 18-Jan-2020)

	Ref	Expression
Hypotheses	uhgrf.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	uhgrf.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
Assertion	uhgrss	$$⊢\left(G\in \mathrm{UHGraph}\wedge F\in \mathrm{dom}E\right)\to E\left(F\right)\subseteq V$$

Proof

Step	Hyp	Ref	Expression
1		uhgrf.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		uhgrf.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
3	1 2	uhgrf	$$⊢G\in \mathrm{UHGraph}\to E:\mathrm{dom}E⟶𝒫V\setminus \left\{\varnothing \right\}$$
4	3	ffvelcdmda	$$⊢\left(G\in \mathrm{UHGraph}\wedge F\in \mathrm{dom}E\right)\to E\left(F\right)\in \left(𝒫V\setminus \left\{\varnothing \right\}\right)$$
5	4	eldifad	$$⊢\left(G\in \mathrm{UHGraph}\wedge F\in \mathrm{dom}E\right)\to E\left(F\right)\in 𝒫V$$
6	5	elpwid	$$⊢\left(G\in \mathrm{UHGraph}\wedge F\in \mathrm{dom}E\right)\to E\left(F\right)\subseteq V$$