Definition df-subg 17638

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17656), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17651), contains the neutral element of the group (see subg0 17647) and contains the inverses for all of its elements (see subginvcl 17650). (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 17635	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 17469	. . 3 class Grp
4	2	cv 1522	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1522	. . . . . 6 class 𝑠
7		cress 15905	. . . . . 6 class ↾s
8	4, 6, 7	co 6690	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2030	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 15904	. . . . . 6 class Base
11	4, 10	cfv 5926	. . . . 5 class (Base‘𝑤)
12	11	cpw 4191	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2945	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4762	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1523	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  17641