Definition df-umgr 25978

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 e. ( ~P v \ { (/) } ) is used as restriction of the class abstraction, although x e. ~P v would be sufficient (see prprrab 13255 and isumgrs 25991). (Contributed by AV, 24-Nov-2020.)

Assertion

Ref	Expression
df-umgr	|- UMGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }

Distinct variable group:   e, g, v, x

Detailed syntax breakdown of Definition df-umgr

Step	Hyp	Ref	Expression
1		cumgr 25976	. 2 class UMGraph
2		ve	. . . . . . . 8 setvar e
3	2	cv 1482	. . . . . . 7 class e
4	3	cdm 5114	. . . . . 6 class dom e
5		vx	. . . . . . . . . 10 setvar x
6	5	cv 1482	. . . . . . . . 9 class x
7		chash 13117	. . . . . . . . 9 class #
8	6, 7	cfv 5888	. . . . . . . 8 class ( # ` x )
9		c2 11070	. . . . . . . 8 class 2
10	8, 9	wceq 1483	. . . . . . 7 wff ( # ` x ) = 2
11		vv	. . . . . . . . . 10 setvar v
12	11	cv 1482	. . . . . . . . 9 class v
13	12	cpw 4158	. . . . . . . 8 class ~P v
14		c0 3915	. . . . . . . . 9 class (/)
15	14	csn 4177	. . . . . . . 8 class { (/) }
16	13, 15	cdif 3571	. . . . . . 7 class ( ~P v \ { (/) } )
17	10, 5, 16	crab 2916	. . . . . 6 class { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
18	4, 17, 3	wf 5884	. . . . 5 wff e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
19		vg	. . . . . . 7 setvar g
20	19	cv 1482	. . . . . 6 class g
21		ciedg 25875	. . . . . 6 class iEdg
22	20, 21	cfv 5888	. . . . 5 class (iEdg ` g )
23	18, 2, 22	wsbc 3435	. . . 4 wff [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
24		cvtx 25874	. . . . 5 class Vtx
25	20, 24	cfv 5888	. . . 4 class (Vtx ` g )
26	23, 11, 25	wsbc 3435	. . 3 wff [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
27	26, 19	cab 2608	. 2 class { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }
28	1, 27	wceq 1483	1 wff UMGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }

This definition is referenced by:  isumgr  25990