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Definition df-uvtx 26691
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 26690 . 2 class UnivVtx
2 vg . . 3 setvar 𝑔
3 cvv 3414 . . 3 class V
4 vn . . . . . . 7 setvar 𝑛
54cv 1655 . . . . . 6 class 𝑛
62cv 1655 . . . . . . 7 class 𝑔
7 vv . . . . . . . 8 setvar 𝑣
87cv 1655 . . . . . . 7 class 𝑣
9 cnbgr 26636 . . . . . . 7 class NeighbVtx
106, 8, 9co 6910 . . . . . 6 class (𝑔 NeighbVtx 𝑣)
115, 10wcel 2164 . . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12 cvtx 26301 . . . . . . 7 class Vtx
136, 12cfv 6127 . . . . . 6 class (Vtx‘𝑔)
148csn 4399 . . . . . 6 class {𝑣}
1513, 14cdif 3795 . . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
1611, 4, 15wral 3117 . . . 4 wff 𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
1716, 7, 13crab 3121 . . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
182, 3, 17cmpt 4954 . 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
191, 18wceq 1656 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  26692
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