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Definition df-subg 17356
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17374), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17369), contains the neutral element of the group (see subg0 17365) and contains the inverses for all of its elements (see subginvcl 17368). (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 17353 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 17187 . . 3 class Grp
42cv 1473 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1473 . . . . . 6 class 𝑠
7 cress 15638 . . . . . 6 class s
84, 6, 7co 6523 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 1975 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 15637 . . . . . 6 class Base
114, 10cfv 5786 . . . . 5 class (Base‘𝑤)
1211cpw 4103 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 2895 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp}
142, 3, 13cmpt 4633 . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
151, 14wceq 1474 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Colors of variables: wff setvar class
This definition is referenced by:  issubg  17359
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