Definition df-usgra 23088

Description: Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph <. V , E >. where E is an injective (one-to-one) function into subsets of V of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)

Assertion

Ref	Expression
df-usgra	|- USGrph = { <. v , e >. | e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }

Distinct variable group:   v, e, x

Detailed syntax breakdown of Definition df-usgra

Step	Hyp	Ref	Expression
1		cusg 23086	. 2 class USGrph
2		ve	. . . . . 6 setvar e
3	2	cv 1361	. . . . 5 class e
4	3	cdm 4827	. . . 4 class dom e
5		vx	. . . . . . . 8 setvar x
6	5	cv 1361	. . . . . . 7 class x
7		chash 12086	. . . . . . 7 class #
8	6, 7	cfv 5406	. . . . . 6 class ( # ` x )
9		c2 10358	. . . . . 6 class 2
10	8, 9	wceq 1362	. . . . 5 wff ( # ` x ) = 2
11		vv	. . . . . . . 8 setvar v
12	11	cv 1361	. . . . . . 7 class v
13	12	cpw 3848	. . . . . 6 class ~P v
14		c0 3625	. . . . . . 7 class (/)
15	14	csn 3865	. . . . . 6 class { (/) }
16	13, 15	cdif 3313	. . . . 5 class ( ~P v \ { (/) } )
17	10, 5, 16	crab 2709	. . . 4 class { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
18	4, 17, 3	wf1 5403	. . 3 wff e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
19	18, 11, 2	copab 4337	. 2 class { <. v , e >. | e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }
20	1, 19	wceq 1362	1 wff USGrph = { <. v , e >. | e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }

This definition is referenced by:  relusgra  23090  isusgra  23094