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Definition df-subg 17638
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17656), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17651), contains the neutral element of the group (see subg0 17647) and contains the inverses for all of its elements (see subginvcl 17650). (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 17635 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 17469 . . 3 class Grp
42cv 1522 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1522 . . . . . 6 class 𝑠
7 cress 15905 . . . . . 6 class s
84, 6, 7co 6690 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 2030 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 15904 . . . . . 6 class Base
114, 10cfv 5926 . . . . 5 class (Base‘𝑤)
1211cpw 4191 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 2945 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp}
142, 3, 13cmpt 4762 . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
151, 14wceq 1523 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Colors of variables: wff setvar class
This definition is referenced by:  issubg  17641
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