Definition df-subg 19047

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19065), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19060), contains the neutral element of the group (see subg0 19056) and contains the inverses for all of its elements (see subginvcl 19059). (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 19044	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18860	. . 3 class Grp
4	2	cv 1532	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1532	. . . . . 6 class 𝑠
7		cress 17179	. . . . . 6 class ↾s
8	4, 6, 7	co 7404	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2098	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17150	. . . . . 6 class Base
11	4, 10	cfv 6536	. . . . 5 class (Base‘𝑤)
12	11	cpw 4597	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3426	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5224	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1533	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19050