Definition df-subg 18667

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 18685), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 18680), contains the neutral element of the group (see subg0 18676) and contains the inverses for all of its elements (see subginvcl 18679). (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 18664	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18492	. . 3 class Grp
4	2	cv 1538	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1538	. . . . . 6 class 𝑠
7		cress 16867	. . . . . 6 class ↾s
8	4, 6, 7	co 7255	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2108	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 16840	. . . . . 6 class Base
11	4, 10	cfv 6418	. . . . 5 class (Base‘𝑤)
12	11	cpw 4530	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3067	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5153	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1539	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  18670