Definition df-usgr 26046

Description: Define the class of all undirected simple graphs (without loops). An undirected simple graph is a special undirected simple pseudograph (see usgruspgr 26073), consisting of a set v (of "vertices") and an injective (one-to-one) function e (representing (indexed) "edges") into subsets of v of cardinality two, representing the two vertices incident to the edge. In contrast to an undirected simple pseudograph, an undirected simple graph has no loops (edges connecting a vertex with itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)

Assertion

Ref	Expression
df-usgr	|- USGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }

Distinct variable group:   e, g, v, x

Detailed syntax breakdown of Definition df-usgr

Step	Hyp	Ref	Expression
1		cusgr 26044	. 2 class USGraph
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	wf1 5885	. . . . 5 wff e : dom e -1-1-> { 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 -1-1-> { 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 -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
27	26, 19	cab 2608	. 2 class { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }
28	1, 27	wceq 1483	1 wff USGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }

This definition is referenced by:  isusgr  26048