Definition df-usgra 25058

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 25055	. 2 class USGrph
2		ve	. . . . . 6 setvar e
3	2	cv 1436	. . . . 5 class e
4	3	cdm 4853	. . . 4 class dom e
5		vx	. . . . . . . 8 setvar x
6	5	cv 1436	. . . . . . 7 class x
7		chash 12521	. . . . . . 7 class #
8	6, 7	cfv 5601	. . . . . 6 class ( # ` x )
9		c2 10666	. . . . . 6 class 2
10	8, 9	wceq 1437	. . . . 5 wff ( # ` x ) = 2
11		vv	. . . . . . . 8 setvar v
12	11	cv 1436	. . . . . . 7 class v
13	12	cpw 3981	. . . . . 6 class ~P v
14		c0 3761	. . . . . . 7 class (/)
15	14	csn 3998	. . . . . 6 class { (/) }
16	13, 15	cdif 3433	. . . . 5 class ( ~P v \ { (/) } )
17	10, 5, 16	crab 2775	. . . 4 class { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
18	4, 17, 3	wf1 5598	. . 3 wff e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
19	18, 11, 2	copab 4481	. 2 class { <. v , e >. | e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }
20	1, 19	wceq 1437	1 wff USGrph = { <. v , e >. | e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }

This definition is referenced by:  relusgra  25060  isusgra  25069  usgraop  25075