Definition df-subg 13550

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13569), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13564), contains the neutral element of the group (see subg0 13560) and contains the inverses for all of its elements (see subginvcl 13563). (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 13547	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13376	. . 3 class Grp
4	2	cv 1372	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1372	. . . . . 6 class 𝑠
7		cress 12877	. . . . . 6 class ↾s
8	4, 6, 7	co 5951	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2177	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 12876	. . . . . 6 class Base
11	4, 10	cfv 5276	. . . . 5 class (Base‘𝑤)
12	11	cpw 3617	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2489	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4109	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1373	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  13553  subgex  13556