Definition df-subg 13926

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13945), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13940), contains the neutral element of the group (see subg0 13936) and contains the inverses for all of its elements (see subginvcl 13939). (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 13923	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13758	. . 3 class Grp
4	2	cv 1397	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1397	. . . . . 6 class 𝑠
7		cress 13300	. . . . . 6 class ↾s
8	4, 6, 7	co 6058	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2205	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13299	. . . . . 6 class Base
11	4, 10	cfv 5357	. . . . 5 class (Base‘𝑤)
12	11	cpw 3674	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2526	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4176	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1398	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  13929  subgex  13932