Definition df-subg 13759

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13778), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13773), contains the neutral element of the group (see subg0 13769) and contains the inverses for all of its elements (see subginvcl 13772). (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 13756	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13585	. . 3 class Grp
4	2	cv 1396	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1396	. . . . . 6 class 𝑠
7		cress 13085	. . . . . 6 class ↾s
8	4, 6, 7	co 6018	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2202	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13084	. . . . . 6 class Base
11	4, 10	cfv 5326	. . . . 5 class (Base‘𝑤)
12	11	cpw 3652	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2514	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4150	. 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  13762  subgex  13765