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Definition df-subg 19071
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 19089), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 19084), contains the neutral element of the group (see subg0 19080) and contains the inverses for all of its elements (see subginvcl 19083). (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 19068 . 2 class SubGrp
2 vw . . 3 setvar 𝑤
3 cgrp 18883 . . 3 class Grp
42cv 1533 . . . . . 6 class 𝑤
5 vs . . . . . . 7 setvar 𝑠
65cv 1533 . . . . . 6 class 𝑠
7 cress 17202 . . . . . 6 class s
84, 6, 7co 7414 . . . . 5 class (𝑤s 𝑠)
98, 3wcel 2099 . . . 4 wff (𝑤s 𝑠) ∈ Grp
10 cbs 17173 . . . . . 6 class Base
114, 10cfv 6542 . . . . 5 class (Base‘𝑤)
1211cpw 4598 . . . 4 class 𝒫 (Base‘𝑤)
139, 5, 12crab 3428 . . 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 1534 1 wff SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤s 𝑠) ∈ Grp})
Colors of variables: wff setvar class
This definition is referenced by:  issubg  19074
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