Definition df-subg 19020

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19038), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19033), contains the neutral element of the group (see subg0 19029) and contains the inverses for all of its elements (see subginvcl 19032). (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 19017	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18830	. . 3 class Grp
4	2	cv 1539	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1539	. . . . . 6 class 𝑠
7		cress 17159	. . . . . 6 class ↾s
8	4, 6, 7	co 7353	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2109	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17138	. . . . . 6 class Base
11	4, 10	cfv 6486	. . . . 5 class (Base‘𝑤)
12	11	cpw 4553	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3396	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5176	. 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  19023