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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 14300 and isumgrs 27833). (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 27818 | . 2 class UMGraph | |
2 | ve | . . . . . . . 8 setvar 𝑒 | |
3 | 2 | cv 1541 | . . . . . . 7 class 𝑒 |
4 | 3 | cdm 5631 | . . . . . 6 class dom 𝑒 |
5 | vx | . . . . . . . . . 10 setvar 𝑥 | |
6 | 5 | cv 1541 | . . . . . . . . 9 class 𝑥 |
7 | chash 14158 | . . . . . . . . 9 class ♯ | |
8 | 6, 7 | cfv 6492 | . . . . . . . 8 class (♯‘𝑥) |
9 | c2 12142 | . . . . . . . 8 class 2 | |
10 | 8, 9 | wceq 1542 | . . . . . . 7 wff (♯‘𝑥) = 2 |
11 | vv | . . . . . . . . . 10 setvar 𝑣 | |
12 | 11 | cv 1541 | . . . . . . . . 9 class 𝑣 |
13 | 12 | cpw 4559 | . . . . . . . 8 class 𝒫 𝑣 |
14 | c0 4281 | . . . . . . . . 9 class ∅ | |
15 | 14 | csn 4585 | . . . . . . . 8 class {∅} |
16 | 13, 15 | cdif 3906 | . . . . . . 7 class (𝒫 𝑣 ∖ {∅}) |
17 | 10, 5, 16 | crab 3406 | . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
18 | 4, 17, 3 | wf 6488 | . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
19 | vg | . . . . . . 7 setvar 𝑔 | |
20 | 19 | cv 1541 | . . . . . 6 class 𝑔 |
21 | ciedg 27734 | . . . . . 6 class iEdg | |
22 | 20, 21 | cfv 6492 | . . . . 5 class (iEdg‘𝑔) |
23 | 18, 2, 22 | wsbc 3738 | . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
24 | cvtx 27733 | . . . . 5 class Vtx | |
25 | 20, 24 | cfv 6492 | . . . 4 class (Vtx‘𝑔) |
26 | 23, 11, 25 | wsbc 3738 | . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
27 | 26, 19 | cab 2715 | . 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} |
28 | 1, 27 | wceq 1542 | 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 27832 |
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