Definition df-uspgr 40380

Description: Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph (see uspgrupgr 40406) or a special undirected simple hypergraph (see uspgrushgr 40405), 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 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

Distinct variable group:   𝑒,𝑔,𝑣,𝑥

Detailed syntax breakdown of Definition df-uspgr

Step	Hyp	Ref	Expression
1		cuspgr 40378	. 2 class USPGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1474	. . . . . . 7 class 𝑒
4	3	cdm 5038	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1474	. . . . . . . . 9 class 𝑥
7		chash 12979	. . . . . . . . 9 class #
8	6, 7	cfv 5804	. . . . . . . 8 class (#‘𝑥)
9		c2 10947	. . . . . . . 8 class 2
10		cle 9954	. . . . . . . 8 class ≤
11	8, 9, 10	wbr 4583	. . . . . . 7 wff (#‘𝑥) ≤ 2
12		vv	. . . . . . . . . 10 setvar 𝑣
13	12	cv 1474	. . . . . . . . 9 class 𝑣
14	13	cpw 4108	. . . . . . . 8 class 𝒫 𝑣
15		c0 3874	. . . . . . . . 9 class ∅
16	15	csn 4125	. . . . . . . 8 class {∅}
17	14, 16	cdif 3537	. . . . . . 7 class (𝒫 𝑣 ∖ {∅})
18	11, 5, 17	crab 2900	. . . . . 6 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
19	4, 18, 3	wf1 5801	. . . . 5 wff 𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
20		vg	. . . . . . 7 setvar 𝑔
21	20	cv 1474	. . . . . 6 class 𝑔
22		ciedg 25674	. . . . . 6 class iEdg
23	21, 22	cfv 5804	. . . . 5 class (iEdg‘𝑔)
24	19, 2, 23	wsbc 3402	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
25		cvtx 25673	. . . . 5 class Vtx
26	21, 25	cfv 5804	. . . 4 class (Vtx‘𝑔)
27	24, 12, 26	wsbc 3402	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
28	27, 20	cab 2596	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
29	1, 28	wceq 1475	1 wff USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

This definition is referenced by:  isuspgr  40382