Definition df-umgr 29114

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 14508 and isumgrs 29127). (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 29112	. 2 class UMGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1535	. . . . . . 7 class 𝑒
4	3	cdm 5688	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1535	. . . . . . . . 9 class 𝑥
7		chash 14365	. . . . . . . . 9 class ♯
8	6, 7	cfv 6562	. . . . . . . 8 class (♯‘𝑥)
9		c2 12318	. . . . . . . 8 class 2
10	8, 9	wceq 1536	. . . . . . 7 wff (♯‘𝑥) = 2
11		vv	. . . . . . . . . 10 setvar 𝑣
12	11	cv 1535	. . . . . . . . 9 class 𝑣
13	12	cpw 4604	. . . . . . . 8 class 𝒫 𝑣
14		c0 4338	. . . . . . . . 9 class ∅
15	14	csn 4630	. . . . . . . 8 class {∅}
16	13, 15	cdif 3959	. . . . . . 7 class (𝒫 𝑣 ∖ {∅})
17	10, 5, 16	crab 3432	. . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
18	4, 17, 3	wf 6558	. . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19		vg	. . . . . . 7 setvar 𝑔
20	19	cv 1535	. . . . . 6 class 𝑔
21		ciedg 29028	. . . . . 6 class iEdg
22	20, 21	cfv 6562	. . . . 5 class (iEdg‘𝑔)
23	18, 2, 22	wsbc 3790	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24		cvtx 29027	. . . . 5 class Vtx
25	20, 24	cfv 6562	. . . 4 class (Vtx‘𝑔)
26	23, 11, 25	wsbc 3790	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
27	26, 19	cab 2711	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
28	1, 27	wceq 1536	1 wff UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}

This definition is referenced by:  isumgr  29126