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Definition df-uvtx 26507
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
StepHypRef Expression
1 cuvtx 26506 . 2 class UnivVtx
2 vg . . 3 setvar 𝑔
3 cvv 3340 . . 3 class V
4 vn . . . . . . 7 setvar 𝑛
54cv 1631 . . . . . 6 class 𝑛
62cv 1631 . . . . . . 7 class 𝑔
7 vv . . . . . . . 8 setvar 𝑣
87cv 1631 . . . . . . 7 class 𝑣
9 cnbgr 26444 . . . . . . 7 class NeighbVtx
106, 8, 9co 6814 . . . . . 6 class (𝑔 NeighbVtx 𝑣)
115, 10wcel 2139 . . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12 cvtx 26094 . . . . . . 7 class Vtx
136, 12cfv 6049 . . . . . 6 class (Vtx‘𝑔)
148csn 4321 . . . . . 6 class {𝑣}
1513, 14cdif 3712 . . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
1611, 4, 15wral 3050 . . . 4 wff 𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
1716, 7, 13crab 3054 . . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
182, 3, 17cmpt 4881 . 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
191, 18wceq 1632 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:  uvtxval  26508  uvtxavalOLD  26509
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