Definition df-subg 19153

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19171), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19166), contains the neutral element of the group (see subg0 19162) and contains the inverses for all of its elements (see subginvcl 19165). (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 19150	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18963	. . 3 class Grp
4	2	cv 1535	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1535	. . . . . 6 class 𝑠
7		cress 17273	. . . . . 6 class ↾s
8	4, 6, 7	co 7430	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2105	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17244	. . . . . 6 class Base
11	4, 10	cfv 6562	. . . . 5 class (Base‘𝑤)
12	11	cpw 4604	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3432	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5230	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1536	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19156