Definition df-subg 19141

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19159), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19154), contains the neutral element of the group (see subg0 19150) and contains the inverses for all of its elements (see subginvcl 19153). (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 19138	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18951	. . 3 class Grp
4	2	cv 1539	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1539	. . . . . 6 class 𝑠
7		cress 17274	. . . . . 6 class ↾s
8	4, 6, 7	co 7431	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2108	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17247	. . . . . 6 class Base
11	4, 10	cfv 6561	. . . . 5 class (Base‘𝑤)
12	11	cpw 4600	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3436	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5225	. 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  19144