Definition df-subg 13728

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13747), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13742), contains the neutral element of the group (see subg0 13738) and contains the inverses for all of its elements (see subginvcl 13741). (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 13725	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13554	. . 3 class Grp
4	2	cv 1394	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1394	. . . . . 6 class 𝑠
7		cress 13054	. . . . . 6 class ↾s
8	4, 6, 7	co 6010	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2200	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13053	. . . . . 6 class Base
11	4, 10	cfv 5321	. . . . 5 class (Base‘𝑤)
12	11	cpw 3649	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2512	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4145	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1395	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  13731  subgex  13734