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Definition df-subg 17414
 Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17432), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17427), contains the neutral element of the group (see subg0 17423) and contains the inverses for all of its elements (see subginvcl 17426). (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 17411 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 17245 . . 3 class Grp
42cv 1474 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1474 . . . . . 6 class 𝑠
7 cress 15696 . . . . . 6 class s
84, 6, 7co 6549 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 1977 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 15695 . . . . . 6 class Base
114, 10cfv 5804 . . . . 5 class (Base‘𝑤)
1211cpw 4108 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 2900 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp}
142, 3, 13cmpt 4643 . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
151, 14wceq 1475 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
 Colors of variables: wff setvar class This definition is referenced by:  issubg  17417
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