Definition df-subg 19146

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19164), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19159), contains the neutral element of the group (see subg0 19155) and contains the inverses for all of its elements (see subginvcl 19158). (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 19143	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18956	. . 3 class Grp
4	2	cv 1558	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1558	. . . . . 6 class 𝑠
7		cress 17247	. . . . . 6 class ↾s
8	4, 6, 7	co 7390	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2141	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17226	. . . . . 6 class Base
11	4, 10	cfv 6515	. . . . 5 class (Base‘𝑤)
12	11	cpw 4554	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3413	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5180	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1559	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19149