Definition df-subg 13376

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13395), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13390), contains the neutral element of the group (see subg0 13386) and contains the inverses for all of its elements (see subginvcl 13389). (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 13373	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13202	. . 3 class Grp
4	2	cv 1363	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1363	. . . . . 6 class 𝑠
7		cress 12704	. . . . . 6 class ↾s
8	4, 6, 7	co 5925	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2167	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 12703	. . . . . 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  13379  subgex  13382