Definition df-subg 13750

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13769), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13764), contains the neutral element of the group (see subg0 13760) and contains the inverses for all of its elements (see subginvcl 13763). (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 13747	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13576	. . 3 class Grp
4	2	cv 1394	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1394	. . . . . 6 class 𝑠
7		cress 13076	. . . . . 6 class ↾s
8	4, 6, 7	co 6013	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2200	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13075	. . . . . 6 class Base
11	4, 10	cfv 5324	. . . . 5 class (Base‘𝑤)
12	11	cpw 3650	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2512	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4148	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1395	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  13753  subgex  13756