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Mirrors > Home > MPE Home > Th. List > df-ushgr | Structured version Visualization version GIF version |
Description: Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function 𝑒 is an injective (one-to-one) function into subsets of the set of vertices 𝑣, representing the (one or more) vertices incident to the edge. This definition corresponds to the definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subset of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are nonempty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by AV, 8-Oct-2020.) |
Ref | Expression |
---|---|
df-ushgr | ⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cushgr 28825 | . 2 class USHGraph | |
2 | ve | . . . . . . . 8 setvar 𝑒 | |
3 | 2 | cv 1532 | . . . . . . 7 class 𝑒 |
4 | 3 | cdm 5669 | . . . . . 6 class dom 𝑒 |
5 | vv | . . . . . . . . 9 setvar 𝑣 | |
6 | 5 | cv 1532 | . . . . . . . 8 class 𝑣 |
7 | 6 | cpw 4597 | . . . . . . 7 class 𝒫 𝑣 |
8 | c0 4317 | . . . . . . . 8 class ∅ | |
9 | 8 | csn 4623 | . . . . . . 7 class {∅} |
10 | 7, 9 | cdif 3940 | . . . . . 6 class (𝒫 𝑣 ∖ {∅}) |
11 | 4, 10, 3 | wf1 6534 | . . . . 5 wff 𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅}) |
12 | vg | . . . . . . 7 setvar 𝑔 | |
13 | 12 | cv 1532 | . . . . . 6 class 𝑔 |
14 | ciedg 28765 | . . . . . 6 class iEdg | |
15 | 13, 14 | cfv 6537 | . . . . 5 class (iEdg‘𝑔) |
16 | 11, 2, 15 | wsbc 3772 | . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅}) |
17 | cvtx 28764 | . . . . 5 class Vtx | |
18 | 13, 17 | cfv 6537 | . . . 4 class (Vtx‘𝑔) |
19 | 16, 5, 18 | wsbc 3772 | . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅}) |
20 | 19, 12 | cab 2703 | . 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})} |
21 | 1, 20 | wceq 1533 | 1 wff USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})} |
Colors of variables: wff setvar class |
This definition is referenced by: isushgr 28829 |
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