Definition df-uhgr

Description: Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set v (of "vertices") and a function e (representing indexed "edges") into the power set of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017) (Revised by AV, 8-Oct-2020)

	Ref	Expression
Assertion	df-uhgr	$$⊢\mathrm{UHGraph}=\left\{g|\underset{\dot{}}{[}\mathrm{Vtx}\left(g\right)/v\underset{\dot{}}{]}\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶𝒫v\setminus \left\{\varnothing \right\}\right\}$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cuhgr	$$class\mathrm{UHGraph}$$
1		vg	$$setvarg$$
2		cvtx	$$class\mathrm{Vtx}$$
3	1	cv	$$setvarg$$
4	3 2	cfv	$$class\mathrm{Vtx}\left(g\right)$$
5		vv	$$setvarv$$
6		ciedg	$$class\mathrm{iEdg}$$
7	3 6	cfv	$$class\mathrm{iEdg}\left(g\right)$$
8		ve	$$setvare$$
9	8	cv	$$setvare$$
10	9	cdm	$$class\mathrm{dom}e$$
11	5	cv	$$setvarv$$
12	11	cpw	$$class𝒫v$$
13		c0	$$class\varnothing$$
14	13	csn	$$class\left\{\varnothing \right\}$$
15	12 14	cdif	$$class\left(𝒫v\setminus \left\{\varnothing \right\}\right)$$
16	10 15 9	wf	$$wffe:\mathrm{dom}e⟶𝒫v\setminus \left\{\varnothing \right\}$$
17	16 8 7	wsbc	$$wff\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶𝒫v\setminus \left\{\varnothing \right\}$$
18	17 5 4	wsbc	$$wff\underset{\dot{}}{[}\mathrm{Vtx}\left(g\right)/v\underset{\dot{}}{]}\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶𝒫v\setminus \left\{\varnothing \right\}$$
19	18 1	cab	$$class\left\{g|\underset{\dot{}}{[}\mathrm{Vtx}\left(g\right)/v\underset{\dot{}}{]}\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶𝒫v\setminus \left\{\varnothing \right\}\right\}$$
20	0 19	wceq	$$wff\mathrm{UHGraph}=\left\{g|\underset{\dot{}}{[}\mathrm{Vtx}\left(g\right)/v\underset{\dot{}}{]}\underset{\dot{}}{[}\mathrm{iEdg}\left(g\right)/e\underset{\dot{}}{]}e:\mathrm{dom}e⟶𝒫v\setminus \left\{\varnothing \right\}\right\}$$