Definition df-subg 19046

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19064), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19059), contains the neutral element of the group (see subg0 19055) and contains the inverses for all of its elements (see subginvcl 19058). (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 19043	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18856	. . 3 class Grp
4	2	cv 1540	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1540	. . . . . 6 class 𝑠
7		cress 17151	. . . . . 6 class ↾s
8	4, 6, 7	co 7355	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2113	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17130	. . . . . 6 class Base
11	4, 10	cfv 6489	. . . . 5 class (Base‘𝑤)
12	11	cpw 4551	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3397	. . 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 1541	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19049