Definition df-subg 19085

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19103), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19098), contains the neutral element of the group (see subg0 19094) and contains the inverses for all of its elements (see subginvcl 19097). (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 19082	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18897	. . 3 class Grp
4	2	cv 1532	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1532	. . . . . 6 class 𝑠
7		cress 17216	. . . . . 6 class ↾s
8	4, 6, 7	co 7426	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2098	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17187	. . . . . 6 class Base
11	4, 10	cfv 6553	. . . . 5 class (Base‘𝑤)
12	11	cpw 4606	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3430	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5235	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1533	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19088