![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 14384 and isumgrs 28110). (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 28095 | . 2 class UMGraph | |
2 | ve | . . . . . . . 8 setvar 𝑒 | |
3 | 2 | cv 1540 | . . . . . . 7 class 𝑒 |
4 | 3 | cdm 5638 | . . . . . 6 class dom 𝑒 |
5 | vx | . . . . . . . . . 10 setvar 𝑥 | |
6 | 5 | cv 1540 | . . . . . . . . 9 class 𝑥 |
7 | chash 14240 | . . . . . . . . 9 class ♯ | |
8 | 6, 7 | cfv 6501 | . . . . . . . 8 class (♯‘𝑥) |
9 | c2 12217 | . . . . . . . 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 4565 | . . . . . . . 8 class 𝒫 𝑣 |
14 | c0 4287 | . . . . . . . . 9 class ∅ | |
15 | 14 | csn 4591 | . . . . . . . 8 class {∅} |
16 | 13, 15 | cdif 3910 | . . . . . . 7 class (𝒫 𝑣 ∖ {∅}) |
17 | 10, 5, 16 | crab 3405 | . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
18 | 4, 17, 3 | wf 6497 | . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
19 | vg | . . . . . . 7 setvar 𝑔 | |
20 | 19 | cv 1540 | . . . . . 6 class 𝑔 |
21 | ciedg 28011 | . . . . . 6 class iEdg | |
22 | 20, 21 | cfv 6501 | . . . . 5 class (iEdg‘𝑔) |
23 | 18, 2, 22 | wsbc 3742 | . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
24 | cvtx 28010 | . . . . 5 class Vtx | |
25 | 20, 24 | cfv 6501 | . . . 4 class (Vtx‘𝑔) |
26 | 23, 11, 25 | wsbc 3742 | . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2} |
27 | 26, 19 | cab 2708 | . 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 28109 |
Copyright terms: Public domain | W3C validator |