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Definition df-umgr 25750
 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 13112 and isumgrs 25762). (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 25748 . 2 class UMGraph
2 ve . . . . . . . 8 setvar 𝑒
32cv 1474 . . . . . . 7 class 𝑒
43cdm 5038 . . . . . 6 class dom 𝑒
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1474 . . . . . . . . 9 class 𝑥
7 chash 12979 . . . . . . . . 9 class #
86, 7cfv 5804 . . . . . . . 8 class (#‘𝑥)
9 c2 10947 . . . . . . . 8 class 2
108, 9wceq 1475 . . . . . . 7 wff (#‘𝑥) = 2
11 vv . . . . . . . . . 10 setvar 𝑣
1211cv 1474 . . . . . . . . 9 class 𝑣
1312cpw 4108 . . . . . . . 8 class 𝒫 𝑣
14 c0 3874 . . . . . . . . 9 class
1514csn 4125 . . . . . . . 8 class {∅}
1613, 15cdif 3537 . . . . . . 7 class (𝒫 𝑣 ∖ {∅})
1710, 5, 16crab 2900 . . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
184, 17, 3wf 5800 . . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
19 vg . . . . . . 7 setvar 𝑔
2019cv 1474 . . . . . 6 class 𝑔
21 ciedg 25674 . . . . . 6 class iEdg
2220, 21cfv 5804 . . . . 5 class (iEdg‘𝑔)
2318, 2, 22wsbc 3402 . . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
24 cvtx 25673 . . . . 5 class Vtx
2520, 24cfv 5804 . . . 4 class (Vtx‘𝑔)
2623, 11, 25wsbc 3402 . . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
2726, 19cab 2596 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}
281, 27wceq 1475 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  25761
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