Definition df-subg 18761

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 18779), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 18774), contains the neutral element of the group (see subg0 18770) and contains the inverses for all of its elements (see subginvcl 18773). (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 18758	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18586	. . 3 class Grp
4	2	cv 1538	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1538	. . . . . 6 class 𝑠
7		cress 16950	. . . . . 6 class ↾s
8	4, 6, 7	co 7284	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2107	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 16921	. . . . . 6 class Base
11	4, 10	cfv 6437	. . . . 5 class (Base‘𝑤)
12	11	cpw 4534	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3069	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5158	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1539	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  18764