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Definition df-subg 19141
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19159), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19154), contains the neutral element of the group (see subg0 19150) and contains the inverses for all of its elements (see subginvcl 19153). (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 19138 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 18951 . . 3 class Grp
42cv 1539 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1539 . . . . . 6 class 𝑠
7 cress 17274 . . . . . 6 class s
84, 6, 7co 7431 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 2108 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 17247 . . . . . 6 class Base
114, 10cfv 6561 . . . . 5 class (Base‘𝑤)
1211cpw 4600 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 3436 . . 3 class {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp}
142, 3, 13cmpt 5225 . 2 class (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
151, 14wceq 1540 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Colors of variables: wff setvar class
This definition is referenced by:  issubg  19144
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