Definition df-subg 19071

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19089), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19084), contains the neutral element of the group (see subg0 19080) and contains the inverses for all of its elements (see subginvcl 19083). (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 19068	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18883	. . 3 class Grp
4	2	cv 1533	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1533	. . . . . 6 class 𝑠
7		cress 17202	. . . . . 6 class ↾s
8	4, 6, 7	co 7414	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2099	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17173	. . . . . 6 class Base
11	4, 10	cfv 6542	. . . . 5 class (Base‘𝑤)
12	11	cpw 4598	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3428	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5225	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1534	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19074