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Definition df-subg 18276
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 18294), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 18289), contains the neutral element of the group (see subg0 18285) and contains the inverses for all of its elements (see subginvcl 18288). (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 18273 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 18103 . . 3 class Grp
42cv 1537 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1537 . . . . . 6 class 𝑠
7 cress 16484 . . . . . 6 class s
84, 6, 7co 7149 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 2115 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 16483 . . . . . 6 class Base
114, 10cfv 6343 . . . . 5 class (Base‘𝑤)
1211cpw 4522 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 3137 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp}
142, 3, 13cmpt 5132 . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
151, 14wceq 1538 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Colors of variables: wff setvar class
This definition is referenced by:  issubg  18279
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