Definition df-subg 12961

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 12980), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 12975), contains the neutral element of the group (see subg0 12971) and contains the inverses for all of its elements (see subginvcl 12974). (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 12958	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 12809	. . 3 class Grp
4	2	cv 1352	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1352	. . . . . 6 class 𝑠
7		cress 12455	. . . . . 6 class ↾s
8	4, 6, 7	co 5872	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2148	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 12454	. . . . . 6 class Base
11	4, 10	cfv 5215	. . . . 5 class (Base‘𝑤)
12	11	cpw 3575	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2459	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4063	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1353	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  12964  subgex  12967