Definition df-subg 19028

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19046), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19041), contains the neutral element of the group (see subg0 19037) and contains the inverses for all of its elements (see subginvcl 19040). (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 19025	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18838	. . 3 class Grp
4	2	cv 1540	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1540	. . . . . 6 class 𝑠
7		cress 17133	. . . . . 6 class ↾s
8	4, 6, 7	co 7341	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2110	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17112	. . . . . 6 class Base
11	4, 10	cfv 6477	. . . . 5 class (Base‘𝑤)
12	11	cpw 4548	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3393	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5170	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1541	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19031