Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-uvtx Structured version   Visualization version   GIF version

Definition df-uvtx 27275
 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 27274 . 2 class UnivVtx
2 vg . . 3 setvar 𝑔
3 cvv 3409 . . 3 class V
4 vn . . . . . . 7 setvar 𝑛
54cv 1537 . . . . . 6 class 𝑛
62cv 1537 . . . . . . 7 class 𝑔
7 vv . . . . . . . 8 setvar 𝑣
87cv 1537 . . . . . . 7 class 𝑣
9 cnbgr 27221 . . . . . . 7 class NeighbVtx
106, 8, 9co 7150 . . . . . 6 class (𝑔 NeighbVtx 𝑣)
115, 10wcel 2111 . . . . 5 wff 𝑛 ∈ (𝑔 NeighbVtx 𝑣)
12 cvtx 26888 . . . . . . 7 class Vtx
136, 12cfv 6335 . . . . . 6 class (Vtx‘𝑔)
148csn 4522 . . . . . 6 class {𝑣}
1513, 14cdif 3855 . . . . 5 class ((Vtx‘𝑔) ∖ {𝑣})
1611, 4, 15wral 3070 . . . 4 wff 𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)
1716, 7, 13crab 3074 . . 3 class {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}
182, 3, 17cmpt 5112 . 2 class (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
191, 18wceq 1538 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  27276
 Copyright terms: Public domain W3C validator