Definition df-subg 18998

Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19016), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19011), contains the neutral element of the group (see subg0 19007) and contains the inverses for all of its elements (see subginvcl 19010). (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 18995	. 2 class SubGrp
2		vw	. . 3 setvar 𝑤
3		cgrp 18816	. . 3 class Grp
4	2	cv 1541	. . . . . 6 class 𝑤
5		vs	. . . . . . 7 setvar 𝑠
6	5	cv 1541	. . . . . 6 class 𝑠
7		cress 17170	. . . . . 6 class ↾s
8	4, 6, 7	co 7406	. . . . 5 class (𝑤 ↾s 𝑠)
9	8, 3	wcel 2107	. . . 4 wff (𝑤 ↾s 𝑠) ∈ Grp
10		cbs 17141	. . . . . 6 class Base
11	4, 10	cfv 6541	. . . . 5 class (Base‘𝑤)
12	11	cpw 4602	. . . 4 class 𝒫 (Base‘𝑤)
13	9, 5, 12	crab 3433	. . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}
14	2, 3, 13	cmpt 5231	. 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  19001