Definition df-upgr 27461

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 27462). (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 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}

Distinct variable group:   𝑒,𝑔,𝑣,𝑥

Detailed syntax breakdown of Definition df-upgr

Step	Hyp	Ref	Expression
1		cupgr 27459	. 2 class UPGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1538	. . . . . . 7 class 𝑒
4	3	cdm 5590	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1538	. . . . . . . . 9 class 𝑥
7		chash 14053	. . . . . . . . 9 class ♯
8	6, 7	cfv 6437	. . . . . . . 8 class (♯‘𝑥)
9		c2 12037	. . . . . . . 8 class 2
10		cle 11019	. . . . . . . 8 class ≤
11	8, 9, 10	wbr 5075	. . . . . . 7 wff (♯‘𝑥) ≤ 2
12		vv	. . . . . . . . . 10 setvar 𝑣
13	12	cv 1538	. . . . . . . . 9 class 𝑣
14	13	cpw 4534	. . . . . . . 8 class 𝒫 𝑣
15		c0 4257	. . . . . . . . 9 class ∅
16	15	csn 4562	. . . . . . . 8 class {∅}
17	14, 16	cdif 3885	. . . . . . 7 class (𝒫 𝑣 ∖ {∅})
18	11, 5, 17	crab 3069	. . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
19	4, 18, 3	wf 6433	. . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
20		vg	. . . . . . 7 setvar 𝑔
21	20	cv 1538	. . . . . 6 class 𝑔
22		ciedg 27376	. . . . . 6 class iEdg
23	21, 22	cfv 6437	. . . . 5 class (iEdg‘𝑔)
24	19, 2, 23	wsbc 3717	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
25		cvtx 27375	. . . . 5 class Vtx
26	21, 25	cfv 6437	. . . 4 class (Vtx‘𝑔)
27	24, 12, 26	wsbc 3717	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
28	27, 20	cab 2716	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}
29	1, 28	wceq 1539	1 wff UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}}

This definition is referenced by:  isupgr  27463