Definition df-subg 13715

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13734), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13729), contains the neutral element of the group (see subg0 13725) and contains the inverses for all of its elements (see subginvcl 13728). (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 13712	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13541	. . 3 class Grp
4	2	cv 1394	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1394	. . . . . 6 class 𝑠
7		cress 13041	. . . . . 6 class ↾s
8	4, 6, 7	co 6007	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2200	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13040	. . . . . 6 class Base
11	4, 10	cfv 5318	. . . . 5 class (Base‘𝑤)
12	11	cpw 3649	. . . 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 4145	. 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  13718  subgex  13721