Definition df-uvtx 29403

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 29402	. 2 class UnivVtx
2		vg	. . 3 setvar 𝑔
3		cvv 3480	. . 3 class V
4		vn	. . . . . . 7 setvar 𝑛
5	4	cv 1539	. . . . . 6 class 𝑛
6	2	cv 1539	. . . . . . 7 class 𝑔
7		vv	. . . . . . . 8 setvar 𝑣
8	7	cv 1539	. . . . . . 7 class 𝑣
9		cnbgr 29349	. . . . . . 7 class NeighbVtx
10	6, 8, 9	co 7431	. . . . . 6 class (𝑔 NeighbVtx 𝑣)
11	5, 10	wcel 2108	. . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12		cvtx 29013	. . . . . . 7 class Vtx
13	6, 12	cfv 6561	. . . . . 6 class (Vtx‘𝑔)
14	8	csn 4626	. . . . . 6 class {𝑣}
15	13, 14	cdif 3948	. . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
16	11, 4, 15	wral 3061	. . . 4 wff ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
17	16, 7, 13	crab 3436	. . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
18	2, 3, 17	cmpt 5225	. 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
19	1, 18	wceq 1540	1 wff UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})

This definition is referenced by:  uvtxval  29404