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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 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 29242 | . 2 class ComplGraph | |
2 | vg | . . . . . 6 setvar 𝑔 | |
3 | 2 | cv 1532 | . . . . 5 class 𝑔 |
4 | cuvtx 29218 | . . . . 5 class UnivVtx | |
5 | 3, 4 | cfv 6553 | . . . 4 class (UnivVtx‘𝑔) |
6 | cvtx 28829 | . . . . 5 class Vtx | |
7 | 3, 6 | cfv 6553 | . . . 4 class (Vtx‘𝑔) |
8 | 5, 7 | wceq 1533 | . . 3 wff (UnivVtx‘𝑔) = (Vtx‘𝑔) |
9 | 8, 2 | cab 2705 | . 2 class {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} |
10 | 1, 9 | wceq 1533 | 1 wff ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)} |
Colors of variables: wff setvar class |
This definition is referenced by: cplgruvtxb 29246 |
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