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Definition df-subg 17943
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17961), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17956), contains the neutral element of the group (see subg0 17952) and contains the inverses for all of its elements (see subginvcl 17955). (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 17940 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 17777 . . 3 class Grp
42cv 1657 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1657 . . . . . 6 class 𝑠
7 cress 16224 . . . . . 6 class s
84, 6, 7co 6906 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 2166 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 16223 . . . . . 6 class Base
114, 10cfv 6124 . . . . 5 class (Base‘𝑤)
1211cpw 4379 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 3122 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp}
142, 3, 13cmpt 4953 . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
151, 14wceq 1658 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Colors of variables: wff setvar class
This definition is referenced by:  issubg  17946
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