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Mirrors > Home > MPE Home > Th. List > df-umgr | Structured version Visualization version GIF version |
Description: Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to a pseudograph, a multigraph has no loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." To provide uniform definitions for all kinds of graphs, 𝑥 ∈ (𝒫 𝑣 ∖ {∅}) is used as restriction of the class abstraction, although 𝑥 ∈ 𝒫 𝑣 would be sufficient (see prprrab 14299 and isumgrs 27845). (Contributed by AV, 24-Nov-2020.) |
Ref | Expression |
---|---|
df-umgr | ⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cumgr 27830 | . 2 class UMGraph | |
2 | ve | . . . . . . . 8 setvar 𝑒 | |
3 | 2 | cv 1540 | . . . . . . 7 class 𝑒 |
4 | 3 | cdm 5630 | . . . . . 6 class dom 𝑒 |
5 | vx | . . . . . . . . . 10 setvar 𝑥 | |
6 | 5 | cv 1540 | . . . . . . . . 9 class 𝑥 |
7 | chash 14157 | . . . . . . . . 9 class ♯ | |
8 | 6, 7 | cfv 6491 | . . . . . . . 8 class (♯‘𝑥) |
9 | c2 12141 | . . . . . . . 8 class 2 | |
10 | 8, 9 | wceq 1541 | . . . . . . 7 wff (♯‘𝑥) = 2 |
11 | vv | . . . . . . . . . 10 setvar 𝑣 | |
12 | 11 | cv 1540 | . . . . . . . . 9 class 𝑣 |
13 | 12 | cpw 4558 | . . . . . . . 8 class 𝒫 𝑣 |
14 | c0 4280 | . . . . . . . . 9 class ∅ | |
15 | 14 | csn 4584 | . . . . . . . 8 class {∅} |
16 | 13, 15 | cdif 3905 | . . . . . . 7 class (𝒫 𝑣 ∖ {∅}) |
17 | 10, 5, 16 | crab 3405 | . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
18 | 4, 17, 3 | wf 6487 | . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
19 | vg | . . . . . . 7 setvar 𝑔 | |
20 | 19 | cv 1540 | . . . . . 6 class 𝑔 |
21 | ciedg 27746 | . . . . . 6 class iEdg | |
22 | 20, 21 | cfv 6491 | . . . . 5 class (iEdg‘𝑔) |
23 | 18, 2, 22 | wsbc 3737 | . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
24 | cvtx 27745 | . . . . 5 class Vtx | |
25 | 20, 24 | cfv 6491 | . . . 4 class (Vtx‘𝑔) |
26 | 23, 11, 25 | wsbc 3737 | . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
27 | 26, 19 | cab 2714 | . 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} |
28 | 1, 27 | wceq 1541 | 1 wff UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} |
Colors of variables: wff setvar class |
This definition is referenced by: isumgr 27844 |
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