Definition df-subg 14033

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 14052), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14047), contains the neutral element of the group (see subg0 14043) and contains the inverses for all of its elements (see subginvcl 14046). (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 14030	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13865	. . 3 class Grp
4	2	cv 1401	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1401	. . . . . 6 class 𝑠
7		cress 13412	. . . . . 6 class ↾s
8	4, 6, 7	co 6088	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2209	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13411	. . . . . 6 class Base
11	4, 10	cfv 5380	. . . . 5 class (Base‘𝑤)
12	11	cpw 3688	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2532	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4193	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1402	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is used by:  issubg  14036  subgex  14039