Definition df-subg 13975

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13994), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13989), contains the neutral element of the group (see subg0 13985) and contains the inverses for all of its elements (see subginvcl 13988). (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 13972	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13807	. . 3 class Grp
4	2	cv 1401	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1401	. . . . . 6 class 𝑠
7		cress 13355	. . . . . 6 class ↾s
8	4, 6, 7	co 6085	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2209	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13354	. . . . . 6 class Base
11	4, 10	cfv 5377	. . . . 5 class (Base‘𝑤)
12	11	cpw 3688	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2532	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4192	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1402	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is used by:  issubg  13978  subgex  13981