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Definition df-subg 17585
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 17603), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 17598), contains the neutral element of the group (see subg0 17594) and contains the inverses for all of its elements (see subginvcl 17597). (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 17582 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 17416 . . 3 class Grp
42cv 1481 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1481 . . . . . 6 class 𝑠
7 cress 15852 . . . . . 6 class s
84, 6, 7co 6647 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 1989 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 15851 . . . . . 6 class Base
114, 10cfv 5886 . . . . 5 class (Base‘𝑤)
1211cpw 4156 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 2915 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp}
142, 3, 13cmpt 4727 . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
151, 14wceq 1482 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Colors of variables: wff setvar class
This definition is referenced by:  issubg  17588
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