Definition df-uvtx 29471

Description: Define the class of all universal vertices (in graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph), or equivalently, if all other vertices are its neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.)

Assertion

Ref	Expression
df-uvtx	⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})

Distinct variable group:   𝑣,𝑔,𝑛

Detailed syntax breakdown of Definition df-uvtx

Step	Hyp	Ref	Expression
1		cuvtx 29470	. 2 class UnivVtx
2		vg	. . 3 setvar 𝑔
3		cvv 3442	. . 3 class V
4		vn	. . . . . . 7 setvar 𝑛
5	4	cv 1541	. . . . . 6 class 𝑛
6	2	cv 1541	. . . . . . 7 class 𝑔
7		vv	. . . . . . . 8 setvar 𝑣
8	7	cv 1541	. . . . . . 7 class 𝑣
9		cnbgr 29417	. . . . . . 7 class NeighbVtx
10	6, 8, 9	co 7368	. . . . . 6 class (𝑔 NeighbVtx 𝑣)
11	5, 10	wcel 2114	. . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12		cvtx 29081	. . . . . . 7 class Vtx
13	6, 12	cfv 6500	. . . . . 6 class (Vtx‘𝑔)
14	8	csn 4582	. . . . . 6 class {𝑣}
15	13, 14	cdif 3900	. . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
16	11, 4, 15	wral 3052	. . . 4 wff ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
17	16, 7, 13	crab 3401	. . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
18	2, 3, 17	cmpt 5181	. 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
19	1, 18	wceq 1542	1 wff UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})

This definition is referenced by:  uvtxval  29472