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

Step	Hyp	Ref	Expression
1		cumgr 27172	. 2 class UMGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1542	. . . . . . 7 class 𝑒
4	3	cdm 5551	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1542	. . . . . . . . 9 class 𝑥
7		chash 13896	. . . . . . . . 9 class ♯
8	6, 7	cfv 6380	. . . . . . . 8 class (♯‘𝑥)
9		c2 11885	. . . . . . . 8 class 2
10	8, 9	wceq 1543	. . . . . . 7 wff (♯‘𝑥) = 2
11		vv	. . . . . . . . . 10 setvar 𝑣
12	11	cv 1542	. . . . . . . . 9 class 𝑣
13	12	cpw 4513	. . . . . . . 8 class 𝒫 𝑣
14		c0 4237	. . . . . . . . 9 class ∅
15	14	csn 4541	. . . . . . . 8 class {∅}
16	13, 15	cdif 3863	. . . . . . 7 class (𝒫 𝑣 ∖ {∅})
17	10, 5, 16	crab 3065	. . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
18	4, 17, 3	wf 6376	. . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19		vg	. . . . . . 7 setvar 𝑔
20	19	cv 1542	. . . . . 6 class 𝑔
21		ciedg 27088	. . . . . 6 class iEdg
22	20, 21	cfv 6380	. . . . 5 class (iEdg‘𝑔)
23	18, 2, 22	wsbc 3694	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24		cvtx 27087	. . . . 5 class Vtx
25	20, 24	cfv 6380	. . . 4 class (Vtx‘𝑔)
26	23, 11, 25	wsbc 3694	. . 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 1543	1 wff UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}

This definition is referenced by:  isumgr  27186