Definition df-usgra 25862

Description: Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph ⟨𝑉, 𝐸⟩ where 𝐸 is an injective (one-to-one) function into subsets of 𝑉 of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)

Assertion

Ref	Expression
df-usgra	⊢ USGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}

Distinct variable group:   𝑣,𝑒,𝑥

Detailed syntax breakdown of Definition df-usgra

Step	Hyp	Ref	Expression
1		cusg 25859	. 2 class USGrph
2		ve	. . . . . 6 setvar 𝑒
3	2	cv 1474	. . . . 5 class 𝑒
4	3	cdm 5038	. . . 4 class dom 𝑒
5		vx	. . . . . . . 8 setvar 𝑥
6	5	cv 1474	. . . . . . 7 class 𝑥
7		chash 12979	. . . . . . 7 class #
8	6, 7	cfv 5804	. . . . . 6 class (#‘𝑥)
9		c2 10947	. . . . . 6 class 2
10	8, 9	wceq 1475	. . . . 5 wff (#‘𝑥) = 2
11		vv	. . . . . . . 8 setvar 𝑣
12	11	cv 1474	. . . . . . 7 class 𝑣
13	12	cpw 4108	. . . . . 6 class 𝒫 𝑣
14		c0 3874	. . . . . . 7 class ∅
15	14	csn 4125	. . . . . 6 class {∅}
16	13, 15	cdif 3537	. . . . 5 class (𝒫 𝑣 ∖ {∅})
17	10, 5, 16	crab 2900	. . . 4 class {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
18	4, 17, 3	wf1 5801	. . 3 wff 𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}
19	18, 11, 2	copab 4642	. 2 class {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}
20	1, 19	wceq 1475	1 wff USGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}

This definition is referenced by:  relusgra  25864  isusgra  25873  usgraop  25879