Definition df-uvtx 29385

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 29384	. 2 class UnivVtx
2		vg	. . 3 setvar 𝑔
3		cvv 3437	. . 3 class V
4		vn	. . . . . . 7 setvar 𝑛
5	4	cv 1540	. . . . . 6 class 𝑛
6	2	cv 1540	. . . . . . 7 class 𝑔
7		vv	. . . . . . . 8 setvar 𝑣
8	7	cv 1540	. . . . . . 7 class 𝑣
9		cnbgr 29331	. . . . . . 7 class NeighbVtx
10	6, 8, 9	co 7355	. . . . . 6 class (𝑔 NeighbVtx 𝑣)
11	5, 10	wcel 2113	. . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12		cvtx 28995	. . . . . . 7 class Vtx
13	6, 12	cfv 6489	. . . . . 6 class (Vtx‘𝑔)
14	8	csn 4577	. . . . . 6 class {𝑣}
15	13, 14	cdif 3895	. . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
16	11, 4, 15	wral 3048	. . . 4 wff ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
17	16, 7, 13	crab 3396	. . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
18	2, 3, 17	cmpt 5176	. 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
19	1, 18	wceq 1541	1 wff UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})

This definition is referenced by:  uvtxval  29386