Definition df-usgren 15911

Description: Define the class of all undirected simple graphs (without loops). An undirected simple graph is a special undirected simple pseudograph, consisting of a set 𝑣 (of "vertices") and an injective (one-to-one) function 𝑒 (representing (indexed) "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to an undirected simple pseudograph, an undirected simple graph has no loops (edges connecting a vertex with itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)

Assertion

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

Distinct variable group:   𝑒,𝑔,𝑣,𝑥

Detailed syntax breakdown of Definition df-usgren

Step	Hyp	Ref	Expression
1		cusgr 15909	. 2 class USGraph
2		ve	. . . . . . . 8 setvar 𝑒
3	2	cv 1372	. . . . . . 7 class 𝑒
4	3	cdm 4694	. . . . . 6 class dom 𝑒
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1372	. . . . . . . 8 class 𝑥
7		c2o 6521	. . . . . . . 8 class 2o
8		cen 6850	. . . . . . . 8 class ≈
9	6, 7, 8	wbr 4060	. . . . . . 7 wff 𝑥 ≈ 2o
10		vv	. . . . . . . . 9 setvar 𝑣
11	10	cv 1372	. . . . . . . 8 class 𝑣
12	11	cpw 3627	. . . . . . 7 class 𝒫 𝑣
13	9, 5, 12	crab 2490	. . . . . 6 class {𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}
14	4, 13, 3	wf1 5288	. . . . 5 wff 𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}
15		vg	. . . . . . 7 setvar 𝑔
16	15	cv 1372	. . . . . 6 class 𝑔
17		ciedg 15773	. . . . . 6 class iEdg
18	16, 17	cfv 5291	. . . . 5 class (iEdg‘𝑔)
19	14, 2, 18	wsbc 3006	. . . 4 wff [(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}
20		cvtx 15772	. . . . 5 class Vtx
21	16, 20	cfv 5291	. . . 4 class (Vtx‘𝑔)
22	19, 10, 21	wsbc 3006	. . 3 wff [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}
23	22, 15	cab 2193	. 2 class {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}}
24	1, 23	wceq 1373	1 wff USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ 𝒫 𝑣 ∣ 𝑥 ≈ 2o}}

This definition is referenced by:  isusgren  15913