Definition df-uspgr 26045

Description: Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph (see uspgrupgr 26071) or a special undirected simple hypergraph (see uspgrushgr 26070), consisting of a set v (of "vertices") and an injective (one-to-one) function e (representing (indexed) "edges") into subsets of v of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a pseudograph, there is at most one edge between two vertices resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)

Assertion

Ref	Expression
df-uspgr	|- USPGraph = { 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-uspgr

Step	Hyp	Ref	Expression
1		cuspgr 26043	. 2 class USPGraph
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		cle 10075	. . . . . . . 8 class <_
11	8, 9, 10	wbr 4653	. . . . . . 7 wff ( # ` x ) <_ 2
12		vv	. . . . . . . . . 10 setvar v
13	12	cv 1482	. . . . . . . . 9 class v
14	13	cpw 4158	. . . . . . . 8 class ~P v
15		c0 3915	. . . . . . . . 9 class (/)
16	15	csn 4177	. . . . . . . 8 class { (/) }
17	14, 16	cdif 3571	. . . . . . 7 class ( ~P v \ { (/) } )
18	11, 5, 17	crab 2916	. . . . . 6 class { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 }
19	4, 18, 3	wf1 5885	. . . . 5 wff e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 }
20		vg	. . . . . . 7 setvar g
21	20	cv 1482	. . . . . 6 class g
22		ciedg 25875	. . . . . 6 class iEdg
23	21, 22	cfv 5888	. . . . 5 class (iEdg ` g )
24	19, 2, 23	wsbc 3435	. . . 4 wff [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 }
25		cvtx 25874	. . . . 5 class Vtx
26	21, 25	cfv 5888	. . . 4 class (Vtx ` g )
27	24, 12, 26	wsbc 3435	. . 3 wff [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 }
28	27, 20	cab 2608	. 2 class { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } }
29	1, 28	wceq 1483	1 wff USPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } }

This definition is referenced by:  isuspgr  26047