Definition df-uvtx 29673

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 29672	. 2 class UnivVtx
2		vg	. . 3 setvar 𝑔
3		cvv 3463	. . 3 class V
4		vn	. . . . . . 7 setvar 𝑛
5	4	cv 1566	. . . . . 6 class 𝑛
6	2	cv 1566	. . . . . . 7 class 𝑔
7		vv	. . . . . . . 8 setvar 𝑣
8	7	cv 1566	. . . . . . 7 class 𝑣
9		cnbgr 29619	. . . . . . 7 class NeighbVtx
10	6, 8, 9	co 7408	. . . . . 6 class (𝑔 NeighbVtx 𝑣)
11	5, 10	wcel 2149	. . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12		cvtx 29283	. . . . . . 7 class Vtx
13	6, 12	cfv 6534	. . . . . 6 class (Vtx‘𝑔)
14	8	csn 4591	. . . . . 6 class {𝑣}
15	13, 14	cdif 3910	. . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
16	11, 4, 15	wral 3085	. . . 4 wff ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
17	16, 7, 13	crab 3423	. . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
18	2, 3, 17	cmpt 5193	. 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
19	1, 18	wceq 1567	1 wff UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})

This definition is referenced by:  uvtxval  29674