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Definition df-umgr 27832
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.)
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 27830 . 2 class UMGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1540 . . . . . . 7 class 𝑒
43cdm 5630 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1540 . . . . . . . . 9 class 𝑥
7 chash 14157 . . . . . . . . 9 class
86, 7cfv 6491 . . . . . . . 8 class (♯‘𝑥)
9 c2 12141 . . . . . . . 8 class 2
108, 9wceq 1541 . . . . . . 7 wff (♯‘𝑥) = 2
11 vv . . . . . . . . . 10 setvar 𝑣
1211cv 1540 . . . . . . . . 9 class 𝑣
1312cpw 4558 . . . . . . . 8 class 𝒫 𝑣
14 c0 4280 . . . . . . . . 9 class
1514csn 4584 . . . . . . . 8 class {∅}
1613, 15cdif 3905 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1710, 5, 16crab 3405 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
184, 17, 3wf 6487 . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19 vg . . . . . . 7 setvar 𝑔
2019cv 1540 . . . . . 6 class 𝑔
21 ciedg 27746 . . . . . 6 class iEdg
2220, 21cfv 6491 . . . . 5 class (iEdg‘𝑔)
2318, 2, 22wsbc 3737 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24 cvtx 27745 . . . . 5 class Vtx
2520, 24cfv 6491 . . . 4 class (Vtx‘𝑔)
2623, 11, 25wsbc 3737 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
2726, 19cab 2714 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
281, 27wceq 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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