Definition df-subg 19334

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19353), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19347), contains the neutral element of the group (see subg0 19343) and contains the inverses for all of its elements (see subginvcl 19346). (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 19331	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 19145	. . 3 class Grp
4	2	cv 1568	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1568	. . . . . 6 class 𝑠
7		cress 17409	. . . . . 6 class ↾s
8	4, 6, 7	co 7420	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2144	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17388	. . . . . 6 class Base
11	4, 10	cfv 6538	. . . . 5 class (Base‘𝑤)
12	11	cpw 4556	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3412	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5185	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1569	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is used by:  issubg  19337