Definition df-upgren 15878

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 15879). (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 ∈ 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 15876	. 2 class UPGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1394	. . . . . . 7 class 𝑒
4	3	cdm 4716	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1394	. . . . . . . . 9 class 𝑥
7		c1o 6545	. . . . . . . . 9 class 1o
8		cen 6875	. . . . . . . . 9 class ≈
9	6, 7, 8	wbr 4082	. . . . . . . 8 wff 𝑥 ≈ 1o
10		c2o 6546	. . . . . . . . 9 class 2o
11	6, 10, 8	wbr 4082	. . . . . . . 8 wff 𝑥 ≈ 2o
12	9, 11	wo 713	. . . . . . 7 wff (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)
13		vv	. . . . . . . . 9 setvar 𝑣
14	13	cv 1394	. . . . . . . 8 class 𝑣
15	14	cpw 3649	. . . . . . 7 class 𝒫 𝑣
16	12, 5, 15	crab 2512	. . . . . 6 class {𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
17	4, 16, 3	wf 5310	. . . . 5 wff 𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
18		vg	. . . . . . 7 setvar 𝑔
19	18	cv 1394	. . . . . 6 class 𝑔
20		ciedg 15799	. . . . . 6 class iEdg
21	19, 20	cfv 5314	. . . . 5 class (iEdg‘𝑔)
22	17, 2, 21	wsbc 3028	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
23		cvtx 15798	. . . . 5 class Vtx
24	19, 23	cfv 5314	. . . 4 class (Vtx‘𝑔)
25	22, 13, 24	wsbc 3028	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
26	25, 18	cab 2215	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}}
27	1, 26	wceq 1395	1 wff UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ 𝒫 𝑣 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}}

This definition is referenced by:  isupgren  15880