Definition df-subg 19163

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19181), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19176), contains the neutral element of the group (see subg0 19172) and contains the inverses for all of its elements (see subginvcl 19175). (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 19160	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18973	. . 3 class Grp
4	2	cv 1536	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1536	. . . . . 6 class 𝑠
7		cress 17287	. . . . . 6 class ↾s
8	4, 6, 7	co 7448	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2108	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17258	. . . . . 6 class Base
11	4, 10	cfv 6573	. . . . 5 class (Base‘𝑤)
12	11	cpw 4622	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3443	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5249	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1537	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19166