MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-umgr Structured version   Visualization version   GIF version

Definition df-umgr 26782
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 13821 and isumgrs 26795). (Contributed by AV, 24-Nov-2020.)
Assertion
Ref Expression
df-umgr UMGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
Distinct variable group:   𝑒,𝑔,𝑣,𝑥

Detailed syntax breakdown of Definition df-umgr
StepHypRef Expression
1 cumgr 26780 . 2 class UMGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1529 . . . . . . 7 class 𝑒
43cdm 5554 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1529 . . . . . . . . 9 class 𝑥
7 chash 13680 . . . . . . . . 9 class
86, 7cfv 6352 . . . . . . . 8 class (♯‘𝑥)
9 c2 11681 . . . . . . . 8 class 2
108, 9wceq 1530 . . . . . . 7 wff (♯‘𝑥) = 2
11 vv . . . . . . . . . 10 setvar 𝑣
1211cv 1529 . . . . . . . . 9 class 𝑣
1312cpw 4542 . . . . . . . 8 class 𝒫 𝑣
14 c0 4295 . . . . . . . . 9 class
1514csn 4564 . . . . . . . 8 class {∅}
1613, 15cdif 3937 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1710, 5, 16crab 3147 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
184, 17, 3wf 6348 . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19 vg . . . . . . 7 setvar 𝑔
2019cv 1529 . . . . . 6 class 𝑔
21 ciedg 26696 . . . . . 6 class iEdg
2220, 21cfv 6352 . . . . 5 class (iEdg‘𝑔)
2318, 2, 22wsbc 3776 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24 cvtx 26695 . . . . 5 class Vtx
2520, 24cfv 6352 . . . 4 class (Vtx‘𝑔)
2623, 11, 25wsbc 3776 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
2726, 19cab 2804 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
281, 27wceq 1530 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  26794
  Copyright terms: Public domain W3C validator