Definition df-subg 13956

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13975), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13970), contains the neutral element of the group (see subg0 13966) and contains the inverses for all of its elements (see subginvcl 13969). (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 13953	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13788	. . 3 class Grp
4	2	cv 1401	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1401	. . . . . 6 class 𝑠
7		cress 13336	. . . . . 6 class ↾s
8	4, 6, 7	co 6079	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2209	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13335	. . . . . 6 class Base
11	4, 10	cfv 5375	. . . . 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 4190	. 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 referenced by:  issubg  13959  subgex  13962