Definition df-umgr 28883

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 14458 and isumgrs 28896). (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 28881	. 2 class UMGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1533	. . . . . . 7 class 𝑒
4	3	cdm 5672	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1533	. . . . . . . . 9 class 𝑥
7		chash 14313	. . . . . . . . 9 class ♯
8	6, 7	cfv 6542	. . . . . . . 8 class (♯‘𝑥)
9		c2 12289	. . . . . . . 8 class 2
10	8, 9	wceq 1534	. . . . . . 7 wff (♯‘𝑥) = 2
11		vv	. . . . . . . . . 10 setvar 𝑣
12	11	cv 1533	. . . . . . . . 9 class 𝑣
13	12	cpw 4598	. . . . . . . 8 class 𝒫 𝑣
14		c0 4318	. . . . . . . . 9 class ∅
15	14	csn 4624	. . . . . . . 8 class {∅}
16	13, 15	cdif 3941	. . . . . . 7 class (𝒫 𝑣 ∖ {∅})
17	10, 5, 16	crab 3427	. . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
18	4, 17, 3	wf 6538	. . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
19		vg	. . . . . . 7 setvar 𝑔
20	19	cv 1533	. . . . . 6 class 𝑔
21		ciedg 28797	. . . . . 6 class iEdg
22	20, 21	cfv 6542	. . . . 5 class (iEdg‘𝑔)
23	18, 2, 22	wsbc 3774	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
24		cvtx 28796	. . . . 5 class Vtx
25	20, 24	cfv 6542	. . . 4 class (Vtx‘𝑔)
26	23, 11, 25	wsbc 3774	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}
27	26, 19	cab 2704	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}
28	1, 27	wceq 1534	1 wff UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}}

This definition is referenced by:  isumgr  28895