Definition df-usgr

Description: Define the class of all undirected simple graphs (without loops). An undirected simple graph is a special undirected simple pseudograph (see usgruspgr ), 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)

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

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cusgr	|- USGraph
1		vg	|- g
2		cvtx	|- Vtx
3	1	cv	|- g
4	3 2	cfv	|- ( Vtx ` g )
5		vv	|- v
6		ciedg	|- iEdg
7	3 6	cfv	|- ( iEdg ` g )
8		ve	|- e
9	8	cv	|- e
10	9	cdm	|- dom e
11		vx	|- x
12	5	cv	|- v
13	12	cpw	|- ~P v
14		c0	|- (/)
15	14	csn	|- { (/) }
16	13 15	cdif	|- ( ~P v \ { (/) } )
17		chash	|- #
18	11	cv	|- x
19	18 17	cfv	|- ( # ` x )
20		c2	|- 2
21	19 20	wceq	|- ( # ` x ) = 2
22	21 11 16	crab	|- { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
23	10 22 9	wf1	|- e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
24	23 8 7	wsbc	|- [. ( iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
25	24 5 4	wsbc	|- [. ( Vtx ` g ) / v ]. [. ( iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 }
26	25 1	cab	|- { g | [. ( Vtx ` g ) / v ]. [. ( iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }
27	0 26	wceq	|- USGraph = { g | [. ( Vtx ` g ) / v ]. [. ( iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) = 2 } }