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Definition df-umgr 27174
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 14039 and isumgrs 27187). (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 27172 . 2 class UMGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1542 . . . . . . 7 class 𝑒
43cdm 5551 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1542 . . . . . . . . 9 class 𝑥
7 chash 13896 . . . . . . . . 9 class
86, 7cfv 6380 . . . . . . . 8 class (♯‘𝑥)
9 c2 11885 . . . . . . . 8 class 2
108, 9wceq 1543 . . . . . . 7 wff (♯‘𝑥) = 2
11 vv . . . . . . . . . 10 setvar 𝑣
1211cv 1542 . . . . . . . . 9 class 𝑣
1312cpw 4513 . . . . . . . 8 class 𝒫 𝑣
14 c0 4237 . . . . . . . . 9 class
1514csn 4541 . . . . . . . 8 class {∅}
1613, 15cdif 3863 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1710, 5, 16crab 3065 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
184, 17, 3wf 6376 . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19 vg . . . . . . 7 setvar 𝑔
2019cv 1542 . . . . . 6 class 𝑔
21 ciedg 27088 . . . . . 6 class iEdg
2220, 21cfv 6380 . . . . 5 class (iEdg‘𝑔)
2318, 2, 22wsbc 3694 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24 cvtx 27087 . . . . 5 class Vtx
2520, 24cfv 6380 . . . 4 class (Vtx‘𝑔)
2623, 11, 25wsbc 3694 . . 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 1543 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  27186
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