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Mirrors > Home > MPE Home > Th. List > df-cplgr | Structured version Visualization version GIF version |
Description: Define the class of all complete "graphs". A class/graph is called complete if every pair of distinct vertices is connected by an edge, i.e. each vertex has all other vertices as neighbors. (Contributed by AV, 24-Oct-2020.) |
Ref | Expression |
---|---|
df-cplgr | ⊢ ComplGraph = {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccplgr 26226 | . 2 class ComplGraph | |
2 | vv | . . . . . 6 setvar 𝑣 | |
3 | 2 | cv 1482 | . . . . 5 class 𝑣 |
4 | vg | . . . . . . 7 setvar 𝑔 | |
5 | 4 | cv 1482 | . . . . . 6 class 𝑔 |
6 | cuvtxa 26225 | . . . . . 6 class UnivVtx | |
7 | 5, 6 | cfv 5888 | . . . . 5 class (UnivVtx‘𝑔) |
8 | 3, 7 | wcel 1990 | . . . 4 wff 𝑣 ∈ (UnivVtx‘𝑔) |
9 | cvtx 25874 | . . . . 5 class Vtx | |
10 | 5, 9 | cfv 5888 | . . . 4 class (Vtx‘𝑔) |
11 | 8, 2, 10 | wral 2912 | . . 3 wff ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔) |
12 | 11, 4 | cab 2608 | . 2 class {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔)} |
13 | 1, 12 | wceq 1483 | 1 wff ComplGraph = {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔)} |
Colors of variables: wff setvar class |
This definition is referenced by: iscplgr 26310 |
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