Definition df-subg 13780

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13799), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13794), contains the neutral element of the group (see subg0 13790) and contains the inverses for all of its elements (see subginvcl 13793). (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 13777	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13606	. . 3 class Grp
4	2	cv 1396	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1396	. . . . . 6 class 𝑠
7		cress 13106	. . . . . 6 class ↾s
8	4, 6, 7	co 6023	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2201	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13105	. . . . . 6 class Base
11	4, 10	cfv 5328	. . . . 5 class (Base‘𝑤)
12	11	cpw 3653	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2513	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4151	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1397	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  13783  subgex  13786