Definition df-subg 13908

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13927), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13922), contains the neutral element of the group (see subg0 13918) and contains the inverses for all of its elements (see subginvcl 13921). (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 13905	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13734	. . 3 class Grp
4	2	cv 1397	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1397	. . . . . 6 class 𝑠
7		cress 13234	. . . . . 6 class ↾s
8	4, 6, 7	co 6052	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2205	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13233	. . . . . 6 class Base
11	4, 10	cfv 5354	. . . . 5 class (Base‘𝑤)
12	11	cpw 3671	. . . 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 4173	. 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  13911  subgex  13914