Definition df-subg 19188

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19207), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19201), contains the neutral element of the group (see subg0 19197) and contains the inverses for all of its elements (see subginvcl 19200). (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 19185	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18999	. . 3 class Grp
4	2	cv 1567	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1567	. . . . . 6 class 𝑠
7		cress 17289	. . . . . 6 class ↾s
8	4, 6, 7	co 7410	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2141	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17268	. . . . . 6 class Base
11	4, 10	cfv 6536	. . . . 5 class (Base‘𝑤)
12	11	cpw 4561	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3414	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5191	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1568	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19191