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Definition df-subg 17789
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17807), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17802), contains the neutral element of the group (see subg0 17798) and contains the inverses for all of its elements (see subginvcl 17801). (Contributed by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
df-subg SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Distinct variable group:   𝑤,𝑠

Detailed syntax breakdown of Definition df-subg
StepHypRef Expression
1 csubg 17786 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 17623 . . 3 class Grp
42cv 1636 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1636 . . . . . 6 class 𝑠
7 cress 16065 . . . . . 6 class s
84, 6, 7co 6870 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 2156 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 16064 . . . . . 6 class Base
114, 10cfv 6097 . . . . 5 class (Base‘𝑤)
1211cpw 4351 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 3100 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp}
142, 3, 13cmpt 4923 . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
151, 14wceq 1637 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Colors of variables: wff setvar class
This definition is referenced by:  issubg  17792
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