Definition df-subg 19214

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19233), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19227), contains the neutral element of the group (see subg0 19223) and contains the inverses for all of its elements (see subginvcl 19226). (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 19211	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 19025	. . 3 class Grp
4	2	cv 1569	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1569	. . . . . 6 class 𝑠
7		cress 17315	. . . . . 6 class ↾s
8	4, 6, 7	co 7423	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2146	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17294	. . . . . 6 class Base
11	4, 10	cfv 6543	. . . . 5 class (Base‘𝑤)
12	11	cpw 4567	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3419	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5197	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1570	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is used by:  issubg  19217