Definition df-subg 13062

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13081), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13076), contains the neutral element of the group (see subg0 13072) and contains the inverses for all of its elements (see subginvcl 13075). (Contributed by Mario Carneiro, 2-Dec-2014.)

Assertion

Ref	Expression
df-subg	⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

Distinct variable group:   𝑤,𝑠

Detailed syntax breakdown of Definition df-subg

Step	Hyp	Ref	Expression
1		csubg 13059	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 12899	. . 3 class Grp
4	2	cv 1362	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1362	. . . . . 6 class 𝑠
7		cress 12477	. . . . . 6 class ↾s
8	4, 6, 7	co 5888	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2158	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 12476	. . . . . 6 class Base
11	4, 10	cfv 5228	. . . . 5 class (Base‘𝑤)
12	11	cpw 3587	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2469	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4076	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1363	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  13065  subgex  13068