Definition df-upgr 25977

Description: Define the class of all undirected pseudographs. An (undirected) pseudograph consists of a set v (of "vertices") and a 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. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop." (in contrast to a multigraph, see df-umgr 25978). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.)

Assertion

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

Distinct variable group:   e, g, v, x

Detailed syntax breakdown of Definition df-upgr

Step	Hyp	Ref	Expression
1		cupgr 25975	. 2 class UPGraph
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	wf 5884	. . . . 5 wff e : dom e --> { 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 --> { 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 --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 }
28	27, 20	cab 2608	. 2 class { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } }
29	1, 28	wceq 1483	1 wff UPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ( ~P v \ { (/) } ) | ( # ` x ) <_ 2 } }

This definition is referenced by:  isupgr  25979