Definition df-subg 19180

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19199), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19193), contains the neutral element of the group (see subg0 19189) and contains the inverses for all of its elements (see subginvcl 19192). (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 19177	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18990	. . 3 class Grp
4	2	cv 1562	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1562	. . . . . 6 class 𝑠
7		cress 17280	. . . . . 6 class ↾s
8	4, 6, 7	co 7400	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2145	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17259	. . . . . 6 class Base
11	4, 10	cfv 6525	. . . . 5 class (Base‘𝑤)
12	11	cpw 4558	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3417	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5186	. 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
15	1, 14	wceq 1563	1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})

This definition is referenced by:  issubg  19183