Definition df-subg 19055

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19073), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19068), contains the neutral element of the group (see subg0 19064) and contains the inverses for all of its elements (see subginvcl 19067). (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 19052	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18865	. . 3 class Grp
4	2	cv 1539	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1539	. . . . . 6 class 𝑠
7		cress 17200	. . . . . 6 class ↾s
8	4, 6, 7	co 7387	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2109	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17179	. . . . . 6 class Base
11	4, 10	cfv 6511	. . . . 5 class (Base‘𝑤)
12	11	cpw 4563	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3405	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5188	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1540	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19058