Definition df-subg 13326

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13345), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13340), contains the neutral element of the group (see subg0 13336) and contains the inverses for all of its elements (see subginvcl 13339). (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 13323	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13158	. . 3 class Grp
4	2	cv 1363	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1363	. . . . . 6 class 𝑠
7		cress 12690	. . . . . 6 class ↾s
8	4, 6, 7	co 5923	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2167	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 12689	. . . . . 6 class Base
11	4, 10	cfv 5259	. . . . 5 class (Base‘𝑤)
12	11	cpw 3606	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2479	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4095	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1364	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  13329  subgex  13332