Theorem uhgriedg0edg0

Description: A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020) (Proof shortened by AV, 8-Dec-2021)

	Ref	Expression
Assertion	uhgriedg0edg0	$$⊢G\in \mathrm{UHGraph}\to \left(\mathrm{Edg}\left(G\right)=\varnothing \leftrightarrow \mathrm{iEdg}\left(G\right)=\varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{iEdg}\left(G\right)=\mathrm{iEdg}\left(G\right)$$
2	1	uhgrfun	$$⊢G\in \mathrm{UHGraph}\to \mathrm{Fun}\mathrm{iEdg}\left(G\right)$$
3		eqid	$$⊢\mathrm{Edg}\left(G\right)=\mathrm{Edg}\left(G\right)$$
4	1 3	edg0iedg0	$$⊢\mathrm{Fun}\mathrm{iEdg}\left(G\right)\to \left(\mathrm{Edg}\left(G\right)=\varnothing \leftrightarrow \mathrm{iEdg}\left(G\right)=\varnothing \right)$$
5	2 4	syl	$$⊢G\in \mathrm{UHGraph}\to \left(\mathrm{Edg}\left(G\right)=\varnothing \leftrightarrow \mathrm{iEdg}\left(G\right)=\varnothing \right)$$