Definition df-upgren 15739

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-umgren 15740). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.) (Revised by Jim Kingdon, 3-Jan-2026.)

Assertion

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

Distinct variable group:   e, g, v, x

Detailed syntax breakdown of Definition df-upgren

Step	Hyp	Ref	Expression
1		cupgr 15737	. 2 class UPGraph
2		ve	. . . . . . . 8 setvar e
3	2	cv 1372	. . . . . . 7 class e
4	3	cdm 4680	. . . . . 6 class dom e
5		vx	. . . . . . . . . 10 setvar x
6	5	cv 1372	. . . . . . . . 9 class x
7		c1o 6505	. . . . . . . . 9 class 1o
8		cen 6835	. . . . . . . . 9 class ~~
9	6, 7, 8	wbr 4048	. . . . . . . 8 wff x ~~ 1o
10		c2o 6506	. . . . . . . . 9 class 2o
11	6, 10, 8	wbr 4048	. . . . . . . 8 wff x ~~ 2o
12	9, 11	wo 710	. . . . . . 7 wff ( x ~~ 1o \/ x ~~ 2o )
13		vv	. . . . . . . . 9 setvar v
14	13	cv 1372	. . . . . . . 8 class v
15	14	cpw 3618	. . . . . . 7 class ~P v
16	12, 5, 15	crab 2489	. . . . . 6 class { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) }
17	4, 16, 3	wf 5273	. . . . 5 wff e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) }
18		vg	. . . . . . 7 setvar g
19	18	cv 1372	. . . . . 6 class g
20		ciedg 15662	. . . . . 6 class iEdg
21	19, 20	cfv 5277	. . . . 5 class (iEdg ` g )
22	17, 2, 21	wsbc 3000	. . . 4 wff [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) }
23		cvtx 15661	. . . . 5 class Vtx
24	19, 23	cfv 5277	. . . 4 class (Vtx ` g )
25	22, 13, 24	wsbc 3000	. . 3 wff [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) }
26	25, 18	cab 2192	. 2 class { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } }
27	1, 26	wceq 1373	1 wff UPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } }

This definition is referenced by:  isupgren  15741