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Definition df-umgr 26380
 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 13543 and isumgrs 26393). (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 26378 . 2 class UMGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1657 . . . . . . 7 class 𝑒
43cdm 5341 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1657 . . . . . . . . 9 class 𝑥
7 chash 13409 . . . . . . . . 9 class
86, 7cfv 6122 . . . . . . . 8 class (♯‘𝑥)
9 c2 11405 . . . . . . . 8 class 2
108, 9wceq 1658 . . . . . . 7 wff (♯‘𝑥) = 2
11 vv . . . . . . . . . 10 setvar 𝑣
1211cv 1657 . . . . . . . . 9 class 𝑣
1312cpw 4377 . . . . . . . 8 class 𝒫 𝑣
14 c0 4143 . . . . . . . . 9 class
1514csn 4396 . . . . . . . 8 class {∅}
1613, 15cdif 3794 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1710, 5, 16crab 3120 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
184, 17, 3wf 6118 . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19 vg . . . . . . 7 setvar 𝑔
2019cv 1657 . . . . . 6 class 𝑔
21 ciedg 26294 . . . . . 6 class iEdg
2220, 21cfv 6122 . . . . 5 class (iEdg‘𝑔)
2318, 2, 22wsbc 3661 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24 cvtx 26293 . . . . 5 class Vtx
2520, 24cfv 6122 . . . 4 class (Vtx‘𝑔)
2623, 11, 25wsbc 3661 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
2726, 19cab 2810 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
281, 27wceq 1658 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  26392
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