Definition df-subg 12983

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13002), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 12997), contains the neutral element of the group (see subg0 12993) and contains the inverses for all of its elements (see subginvcl 12996). (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 12980	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 12831	. . 3 class Grp
4	2	cv 1352	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1352	. . . . . 6 class 𝑠
7		cress 12457	. . . . . 6 class ↾s
8	4, 6, 7	co 5874	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2148	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 12456	. . . . . 6 class Base
11	4, 10	cfv 5216	. . . . 5 class (Base‘𝑤)
12	11	cpw 3575	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2459	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4064	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1353	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  12986  subgex  12989