Definition df-subg 17414

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17432), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17427), contains the neutral element of the group (see subg0 17423) and contains the inverses for all of its elements (see subginvcl 17426). (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 17411	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 17245	. . 3 class Grp
4	2	cv 1474	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1474	. . . . . 6 class 𝑠
7		cress 15696	. . . . . 6 class ↾s
8	4, 6, 7	co 6549	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 1977	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 15695	. . . . . 6 class Base
11	4, 10	cfv 5804	. . . . 5 class (Base‘𝑤)
12	11	cpw 4108	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2900	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4643	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1475	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  17417