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Definition df-uvtx 29112
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 29111 . 2 class UnivVtx
2 vg . . 3 setvar 𝑔
3 cvv 3466 . . 3 class V
4 vn . . . . . . 7 setvar 𝑛
54cv 1532 . . . . . 6 class 𝑛
62cv 1532 . . . . . . 7 class 𝑔
7 vv . . . . . . . 8 setvar 𝑣
87cv 1532 . . . . . . 7 class 𝑣
9 cnbgr 29058 . . . . . . 7 class NeighbVtx
106, 8, 9co 7401 . . . . . 6 class (𝑔 NeighbVtx 𝑣)
115, 10wcel 2098 . . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12 cvtx 28725 . . . . . . 7 class Vtx
136, 12cfv 6533 . . . . . 6 class (Vtx‘𝑔)
148csn 4620 . . . . . 6 class {𝑣}
1513, 14cdif 3937 . . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
1611, 4, 15wral 3053 . . . 4 wff 𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
1716, 7, 13crab 3424 . . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
182, 3, 17cmpt 5221 . 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
191, 18wceq 1533 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  29113
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