Definition df-subg 18268

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 18286), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 18281), contains the neutral element of the group (see subg0 18277) and contains the inverses for all of its elements (see subginvcl 18280). (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 18265	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18095	. . 3 class Grp
4	2	cv 1537	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1537	. . . . . 6 class 𝑠
7		cress 16476	. . . . . 6 class ↾s
8	4, 6, 7	co 7135	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2111	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 16475	. . . . . 6 class Base
11	4, 10	cfv 6324	. . . . 5 class (Base‘𝑤)
12	11	cpw 4497	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3110	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5110	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1538	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  18271