Definition df-subg 19097

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19115), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19110), contains the neutral element of the group (see subg0 19106) and contains the inverses for all of its elements (see subginvcl 19109). (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 19094	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18907	. . 3 class Grp
4	2	cv 1546	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1546	. . . . . 6 class 𝑠
7		cress 17198	. . . . . 6 class ↾s
8	4, 6, 7	co 7363	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2119	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17177	. . . . . 6 class Base
11	4, 10	cfv 6492	. . . . 5 class (Base‘𝑤)
12	11	cpw 4536	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3392	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5160	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1547	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19100