Definition df-uspgren 15910

Description: Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph or a special undirected simple hypergraph, 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 Jim Kingdon, 15-Jan-2026.)

Assertion

Ref	Expression
df-uspgren	⊢ USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}}

Distinct variable group:   𝑒,𝑔,𝑣,𝑥

Detailed syntax breakdown of Definition df-uspgren

Step	Hyp	Ref	Expression
1		cuspgr 15908	. 2 class USPGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1372	. . . . . . 7 class 𝑒
4	3	cdm 4694	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1372	. . . . . . . . 9 class 𝑥
7		c1o 6520	. . . . . . . . 9 class 1o
8		cen 6850	. . . . . . . . 9 class ≈
9	6, 7, 8	wbr 4060	. . . . . . . 8 wff 𝑥 ≈ 1o
10		c2o 6521	. . . . . . . . 9 class 2o
11	6, 10, 8	wbr 4060	. . . . . . . 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 3627	. . . . . . 7 class 𝒫 𝑣
16	12, 5, 15	crab 2490	. . . . . 6 class {𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
17	4, 16, 3	wf1 5288	. . . . 5 wff 𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
18		vg	. . . . . . 7 setvar 𝑔
19	18	cv 1372	. . . . . 6 class 𝑔
20		ciedg 15773	. . . . . 6 class iEdg
21	19, 20	cfv 5291	. . . . 5 class (iEdg‘𝑔)
22	17, 2, 21	wsbc 3006	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
23		cvtx 15772	. . . . 5 class Vtx
24	19, 23	cfv 5291	. . . 4 class (Vtx‘𝑔)
25	22, 13, 24	wsbc 3006	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
26	25, 18	cab 2193	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}}
27	1, 26	wceq 1373	1 wff USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}}

This definition is referenced by:  isuspgren  15912