Definition df-subg 19065

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19083), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19078), contains the neutral element of the group (see subg0 19074) and contains the inverses for all of its elements (see subginvcl 19077). (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 19062	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18875	. . 3 class Grp
4	2	cv 1541	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1541	. . . . . 6 class 𝑠
7		cress 17169	. . . . . 6 class ↾s
8	4, 6, 7	co 7368	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2114	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17148	. . . . . 6 class Base
11	4, 10	cfv 6500	. . . . 5 class (Base‘𝑤)
12	11	cpw 4556	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3401	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5181	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1542	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19068