Definition df-uhgr 27437

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.)

Assertion

Ref	Expression
df-uhgr	⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}

Distinct variable group:   𝑒,𝑔,𝑣

Detailed syntax breakdown of Definition df-uhgr

Step	Hyp	Ref	Expression
1		cuhgr 27435	. 2 class UHGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1538	. . . . . . 7 class 𝑒
4	3	cdm 5590	. . . . . 6 class dom 𝑒
5		vv	. . . . . . . . 9 setvar 𝑣
6	5	cv 1538	. . . . . . . 8 class 𝑣
7	6	cpw 4534	. . . . . . 7 class 𝒫 𝑣
8		c0 4257	. . . . . . . 8 class ∅
9	8	csn 4562	. . . . . . 7 class {∅}
10	7, 9	cdif 3885	. . . . . 6 class (𝒫 𝑣 ∖ {∅})
11	4, 10, 3	wf 6433	. . . . 5 wff 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})
12		vg	. . . . . . 7 setvar 𝑔
13	12	cv 1538	. . . . . 6 class 𝑔
14		ciedg 27376	. . . . . 6 class iEdg
15	13, 14	cfv 6437	. . . . 5 class (iEdg‘𝑔)
16	11, 2, 15	wsbc 3717	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})
17		cvtx 27375	. . . . 5 class Vtx
18	13, 17	cfv 6437	. . . 4 class (Vtx‘𝑔)
19	16, 5, 18	wsbc 3717	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})
20	19, 12	cab 2716	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
21	1, 20	wceq 1539	1 wff UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}

This definition is referenced by:  isuhgr  27439