Definition df-subg 19104

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19122), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19117), contains the neutral element of the group (see subg0 19113) and contains the inverses for all of its elements (see subginvcl 19116). (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 19101	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18914	. . 3 class Grp
4	2	cv 1539	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1539	. . . . . 6 class 𝑠
7		cress 17249	. . . . . 6 class ↾s
8	4, 6, 7	co 7403	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2108	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17226	. . . . . 6 class Base
11	4, 10	cfv 6530	. . . . 5 class (Base‘𝑤)
12	11	cpw 4575	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3415	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5201	. 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  19107