Definition df-subg 19252

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19271), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19265), contains the neutral element of the group (see subg0 19261) and contains the inverses for all of its elements (see subginvcl 19264). (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 19249	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 19063	. . 3 class Grp
4	2	cv 1569	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1569	. . . . . 6 class 𝑠
7		cress 17328	. . . . . 6 class ↾s
8	4, 6, 7	co 7417	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2145	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17307	. . . . . 6 class Base
11	4, 10	cfv 6537	. . . . 5 class (Base‘𝑤)
12	11	cpw 4560	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3414	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5190	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1570	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is used by:  issubg  19255