Definition df-subg 19099

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19117), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19112), contains the neutral element of the group (see subg0 19108) and contains the inverses for all of its elements (see subginvcl 19111). (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 19096	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18909	. . 3 class Grp
4	2	cv 1541	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1541	. . . . . 6 class 𝑠
7		cress 17200	. . . . . 6 class ↾s
8	4, 6, 7	co 7367	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2114	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17179	. . . . . 6 class Base
11	4, 10	cfv 6499	. . . . 5 class (Base‘𝑤)
12	11	cpw 4542	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3390	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5167	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1542	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19102