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Definition df-umgr 27820
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.)
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 27818 . 2 class UMGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1541 . . . . . . 7 class 𝑒
43cdm 5631 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1541 . . . . . . . . 9 class 𝑥
7 chash 14158 . . . . . . . . 9 class
86, 7cfv 6492 . . . . . . . 8 class (♯‘𝑥)
9 c2 12142 . . . . . . . 8 class 2
108, 9wceq 1542 . . . . . . 7 wff (♯‘𝑥) = 2
11 vv . . . . . . . . . 10 setvar 𝑣
1211cv 1541 . . . . . . . . 9 class 𝑣
1312cpw 4559 . . . . . . . 8 class 𝒫 𝑣
14 c0 4281 . . . . . . . . 9 class
1514csn 4585 . . . . . . . 8 class {∅}
1613, 15cdif 3906 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1710, 5, 16crab 3406 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
184, 17, 3wf 6488 . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19 vg . . . . . . 7 setvar 𝑔
2019cv 1541 . . . . . 6 class 𝑔
21 ciedg 27734 . . . . . 6 class iEdg
2220, 21cfv 6492 . . . . 5 class (iEdg‘𝑔)
2318, 2, 22wsbc 3738 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24 cvtx 27733 . . . . 5 class Vtx
2520, 24cfv 6492 . . . 4 class (Vtx‘𝑔)
2623, 11, 25wsbc 3738 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
2726, 19cab 2715 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
281, 27wceq 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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