Definition df-usgra 23288

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 23286	. 2 class USGrph
2		ve	. . . . . 6 setvar e
3	2	cv 1368	. . . . 5 class e
4	3	cdm 4861	. . . 4 class dom e
5		vx	. . . . . . . 8 setvar x
6	5	cv 1368	. . . . . . 7 class x
7		chash 12124	. . . . . . 7 class #
8	6, 7	cfv 5439	. . . . . 6 class ( # ` x )
9		c2 10392	. . . . . 6 class 2
10	8, 9	wceq 1369	. . . . 5 wff ( # ` x ) = 2
11		vv	. . . . . . . 8 setvar v
12	11	cv 1368	. . . . . . 7 class v
13	12	cpw 3881	. . . . . 6 class ~P v
14		c0 3658	. . . . . . 7 class (/)
15	14	csn 3898	. . . . . 6 class { (/) }
16	13, 15	cdif 3346	. . . . 5 class ( ~P v \ { (/) } )
17	10, 5, 16	crab 2740	. . . 4 class { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
18	4, 17, 3	wf1 5436	. . 3 wff e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
19	18, 11, 2	copab 4370	. 2 class { <. v , e >. | e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }
20	1, 19	wceq 1369	1 wff USGrph = { <. v , e >. | e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }

This definition is referenced by:  relusgra  23290  isusgra  23294