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Definition df-subg 18216
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 18234), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 18229), contains the neutral element of the group (see subg0 18225) and contains the inverses for all of its elements (see subginvcl 18228). (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 18213 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 18043 . . 3 class Grp
42cv 1527 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1527 . . . . . 6 class 𝑠
7 cress 16474 . . . . . 6 class s
84, 6, 7co 7145 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 2105 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 16473 . . . . . 6 class Base
114, 10cfv 6349 . . . . 5 class (Base‘𝑤)
1211cpw 4537 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 3142 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp}
142, 3, 13cmpt 5138 . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
151, 14wceq 1528 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Colors of variables: wff setvar class
This definition is referenced by:  issubg  18219
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