Definition df-subg 18750

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 18768), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 18763), contains the neutral element of the group (see subg0 18759) and contains the inverses for all of its elements (see subginvcl 18762). (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 18747	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18575	. . 3 class Grp
4	2	cv 1541	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1541	. . . . . 6 class 𝑠
7		cress 16939	. . . . . 6 class ↾s
8	4, 6, 7	co 7271	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2110	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 16910	. . . . . 6 class Base
11	4, 10	cfv 6432	. . . . 5 class (Base‘𝑤)
12	11	cpw 4539	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3070	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5162	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1542	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  18753