Definition df-umgr 27862

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 14325 and isumgrs 27875). (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

Step	Hyp	Ref	Expression
1		cumgr 27860	. 2 class UMGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1540	. . . . . . 7 class 𝑒
4	3	cdm 5631	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1540	. . . . . . . . 9 class 𝑥
7		chash 14183	. . . . . . . . 9 class ♯
8	6, 7	cfv 6493	. . . . . . . 8 class (♯‘𝑥)
9		c2 12166	. . . . . . . 8 class 2
10	8, 9	wceq 1541	. . . . . . 7 wff (♯‘𝑥) = 2
11		vv	. . . . . . . . . 10 setvar 𝑣
12	11	cv 1540	. . . . . . . . 9 class 𝑣
13	12	cpw 4558	. . . . . . . 8 class 𝒫 𝑣
14		c0 4280	. . . . . . . . 9 class ∅
15	14	csn 4584	. . . . . . . 8 class {∅}
16	13, 15	cdif 3905	. . . . . . 7 class (𝒫 𝑣 ∖ {∅})
17	10, 5, 16	crab 3405	. . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
18	4, 17, 3	wf 6489	. . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19		vg	. . . . . . 7 setvar 𝑔
20	19	cv 1540	. . . . . 6 class 𝑔
21		ciedg 27776	. . . . . 6 class iEdg
22	20, 21	cfv 6493	. . . . 5 class (iEdg‘𝑔)
23	18, 2, 22	wsbc 3737	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24		cvtx 27775	. . . . 5 class Vtx
25	20, 24	cfv 6493	. . . 4 class (Vtx‘𝑔)
26	23, 11, 25	wsbc 3737	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
27	26, 19	cab 2714	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
28	1, 27	wceq 1541	1 wff UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}

This definition is referenced by:  isumgr  27874