Definition df-uvtx 29418

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 29417	. 2 class UnivVtx
2		vg	. . 3 setvar 𝑔
3		cvv 3478	. . 3 class V
4		vn	. . . . . . 7 setvar 𝑛
5	4	cv 1536	. . . . . 6 class 𝑛
6	2	cv 1536	. . . . . . 7 class 𝑔
7		vv	. . . . . . . 8 setvar 𝑣
8	7	cv 1536	. . . . . . 7 class 𝑣
9		cnbgr 29364	. . . . . . 7 class NeighbVtx
10	6, 8, 9	co 7431	. . . . . 6 class (𝑔 NeighbVtx 𝑣)
11	5, 10	wcel 2106	. . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12		cvtx 29028	. . . . . . 7 class Vtx
13	6, 12	cfv 6563	. . . . . 6 class (Vtx‘𝑔)
14	8	csn 4631	. . . . . 6 class {𝑣}
15	13, 14	cdif 3960	. . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
16	11, 4, 15	wral 3059	. . . 4 wff ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
17	16, 7, 13	crab 3433	. . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
18	2, 3, 17	cmpt 5231	. 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
19	1, 18	wceq 1537	1 wff UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})

This definition is referenced by:  uvtxval  29419