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Definition df-uvtxa 40556
 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 40551 . 2 class UnivVtx
2 vg . . 3 setvar 𝑔
3 cvv 3173 . . 3 class V
4 vn . . . . . . 7 setvar 𝑛
54cv 1474 . . . . . 6 class 𝑛
62cv 1474 . . . . . . 7 class 𝑔
7 vv . . . . . . . 8 setvar 𝑣
87cv 1474 . . . . . . 7 class 𝑣
9 cnbgr 40550 . . . . . . 7 class NeighbVtx
106, 8, 9co 6549 . . . . . 6 class (𝑔 NeighbVtx 𝑣)
115, 10wcel 1977 . . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12 cvtx 25673 . . . . . . 7 class Vtx
136, 12cfv 5804 . . . . . 6 class (Vtx‘𝑔)
148csn 4125 . . . . . 6 class {𝑣}
1513, 14cdif 3537 . . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
1611, 4, 15wral 2896 . . . 4 wff 𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
1716, 7, 13crab 2900 . . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
182, 3, 17cmpt 4643 . 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
191, 18wceq 1475 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  40613
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