Definition df-subg 19295

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19314), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19308), contains the neutral element of the group (see subg0 19304) and contains the inverses for all of its elements (see subginvcl 19307). (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 19292	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 19106	. . 3 class Grp
4	2	cv 1569	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1569	. . . . . 6 class 𝑠
7		cress 17370	. . . . . 6 class ↾s
8	4, 6, 7	co 7408	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2145	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17349	. . . . . 6 class Base
11	4, 10	cfv 6527	. . . . 5 class (Base‘𝑤)
12	11	cpw 4556	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3412	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5185	. 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  19298