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Definition df-uvtxa 26224
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) resp. 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-uvtxa 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-uvtxa
StepHypRef Expression
1 cuvtxa 26219 . 2 class UnivVtx
2 vg . . 3 setvar 𝑔
3 cvv 3198 . . 3 class V
4 vn . . . . . . 7 setvar 𝑛
54cv 1481 . . . . . 6 class 𝑛
62cv 1481 . . . . . . 7 class 𝑔
7 vv . . . . . . . 8 setvar 𝑣
87cv 1481 . . . . . . 7 class 𝑣
9 cnbgr 26218 . . . . . . 7 class NeighbVtx
106, 8, 9co 6647 . . . . . 6 class (𝑔 NeighbVtx 𝑣)
115, 10wcel 1989 . . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12 cvtx 25868 . . . . . . 7 class Vtx
136, 12cfv 5886 . . . . . 6 class (Vtx‘𝑔)
148csn 4175 . . . . . 6 class {𝑣}
1513, 14cdif 3569 . . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
1611, 4, 15wral 2911 . . . 4 wff 𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
1716, 7, 13crab 2915 . . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
182, 3, 17cmpt 4727 . 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
191, 18wceq 1482 1 wff UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
Colors of variables: wff setvar class
This definition is referenced by:  uvtxaval  26281
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