![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 or, in other words, each vertex is a universal vertex. (Contributed by AV, 24-Oct-2020.) (Revised by TA, 15-Feb-2022.) |
Ref | Expression |
---|---|
df-cplgr | ⊢ ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccplgr 26714 | . 2 class ComplGraph | |
2 | vg | . . . . . 6 setvar 𝑔 | |
3 | 2 | cv 1655 | . . . . 5 class 𝑔 |
4 | cuvtx 26690 | . . . . 5 class UnivVtx | |
5 | 3, 4 | cfv 6127 | . . . 4 class (UnivVtx‘𝑔) |
6 | cvtx 26301 | . . . . 5 class Vtx | |
7 | 3, 6 | cfv 6127 | . . . 4 class (Vtx‘𝑔) |
8 | 5, 7 | wceq 1656 | . . 3 wff (UnivVtx‘𝑔) = (Vtx‘𝑔) |
9 | 8, 2 | cab 2811 | . 2 class {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} |
10 | 1, 9 | wceq 1656 | 1 wff ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} |
Colors of variables: wff setvar class |
This definition is referenced by: cplgruvtxb 26718 |
Copyright terms: Public domain | W3C validator |