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Definition df-uvtx 29403
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 29402 . 2 class UnivVtx
2 vg . . 3 setvar 𝑔
3 cvv 3480 . . 3 class V
4 vn . . . . . . 7 setvar 𝑛
54cv 1539 . . . . . 6 class 𝑛
62cv 1539 . . . . . . 7 class 𝑔
7 vv . . . . . . . 8 setvar 𝑣
87cv 1539 . . . . . . 7 class 𝑣
9 cnbgr 29349 . . . . . . 7 class NeighbVtx
106, 8, 9co 7431 . . . . . 6 class (𝑔 NeighbVtx 𝑣)
115, 10wcel 2108 . . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12 cvtx 29013 . . . . . . 7 class Vtx
136, 12cfv 6561 . . . . . 6 class (Vtx‘𝑔)
148csn 4626 . . . . . 6 class {𝑣}
1513, 14cdif 3948 . . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
1611, 4, 15wral 3061 . . . 4 wff 𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
1716, 7, 13crab 3436 . . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
182, 3, 17cmpt 5225 . 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
191, 18wceq 1540 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  29404
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