Definition df-subg 19062

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19080), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19075), contains the neutral element of the group (see subg0 19071) and contains the inverses for all of its elements (see subginvcl 19074). (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 19059	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18872	. . 3 class Grp
4	2	cv 1539	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1539	. . . . . 6 class 𝑠
7		cress 17207	. . . . . 6 class ↾s
8	4, 6, 7	co 7390	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2109	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17186	. . . . . 6 class Base
11	4, 10	cfv 6514	. . . . 5 class (Base‘𝑤)
12	11	cpw 4566	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3408	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5191	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1540	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19065