Definition df-subg 19362

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19381), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19375), contains the neutral element of the group (see subg0 19371) and contains the inverses for all of its elements (see subginvcl 19374). (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 19359	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 19173	. . 3 class Grp
4	2	cv 1569	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1569	. . . . . 6 class 𝑠
7		cress 17437	. . . . . 6 class ↾s
8	4, 6, 7	co 7443	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2147	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17416	. . . . . 6 class Base
11	4, 10	cfv 6558	. . . . 5 class (Base‘𝑤)
12	11	cpw 4560	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3416	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5189	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1570	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is used by:  issubg  19365