Definition df-subg 19057

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19075), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19070), contains the neutral element of the group (see subg0 19066) and contains the inverses for all of its elements (see subginvcl 19069). (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 19054	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18867	. . 3 class Grp
4	2	cv 1541	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1541	. . . . . 6 class 𝑠
7		cress 17161	. . . . . 6 class ↾s
8	4, 6, 7	co 7360	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2114	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17140	. . . . . 6 class Base
11	4, 10	cfv 6493	. . . . 5 class (Base‘𝑤)
12	11	cpw 4555	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3400	. . 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 1542	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19060