Definition df-subg 18276

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 18294), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 18289), contains the neutral element of the group (see subg0 18285) and contains the inverses for all of its elements (see subginvcl 18288). (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 18273	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18103	. . 3 class Grp
4	2	cv 1536	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1536	. . . . . 6 class 𝑠
7		cress 16484	. . . . . 6 class ↾s
8	4, 6, 7	co 7156	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2114	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 16483	. . . . . 6 class Base
11	4, 10	cfv 6355	. . . . 5 class (Base‘𝑤)
12	11	cpw 4539	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3142	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5146	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1537	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  18279