Definition df-subg 13876

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13895), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13890), contains the neutral element of the group (see subg0 13886) and contains the inverses for all of its elements (see subginvcl 13889). (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 13873	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 13702	. . 3 class Grp
4	2	cv 1397	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1397	. . . . . 6 class 𝑠
7		cress 13202	. . . . . 6 class ↾s
8	4, 6, 7	co 6049	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2203	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 13201	. . . . . 6 class Base
11	4, 10	cfv 5351	. . . . 5 class (Base‘𝑤)
12	11	cpw 3668	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 2524	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 4170	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1398	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  13879  subgex  13882