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

Step	Hyp	Ref	Expression
1		cumgr 25748	. 2 class UMGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1474	. . . . . . 7 class 𝑒
4	3	cdm 5038	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1474	. . . . . . . . 9 class 𝑥
7		chash 12979	. . . . . . . . 9 class #
8	6, 7	cfv 5804	. . . . . . . 8 class (#‘𝑥)
9		c2 10947	. . . . . . . 8 class 2
10	8, 9	wceq 1475	. . . . . . 7 wff (#‘𝑥) = 2
11		vv	. . . . . . . . . 10 setvar 𝑣
12	11	cv 1474	. . . . . . . . 9 class 𝑣
13	12	cpw 4108	. . . . . . . 8 class 𝒫 𝑣
14		c0 3874	. . . . . . . . 9 class ∅
15	14	csn 4125	. . . . . . . 8 class {∅}
16	13, 15	cdif 3537	. . . . . . 7 class (𝒫 𝑣 ∖ {∅})
17	10, 5, 16	crab 2900	. . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
18	4, 17, 3	wf 5800	. . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
19		vg	. . . . . . 7 setvar 𝑔
20	19	cv 1474	. . . . . 6 class 𝑔
21		ciedg 25674	. . . . . 6 class iEdg
22	20, 21	cfv 5804	. . . . 5 class (iEdg‘𝑔)
23	18, 2, 22	wsbc 3402	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
24		cvtx 25673	. . . . 5 class Vtx
25	20, 24	cfv 5804	. . . 4 class (Vtx‘𝑔)
26	23, 11, 25	wsbc 3402	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
27	26, 19	cab 2596	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}
28	1, 27	wceq 1475	1 wff UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}

This definition is referenced by:  isumgr  25761