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Definition df-subg 14618
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14636), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14631), contains the neutral element of the group (see subg0 14627) and contains the inverses for all of its elements (see subginvcl 14630). (Contributed by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
df-subg  |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
Distinct variable group:    w, s

Detailed syntax breakdown of Definition df-subg
StepHypRef Expression
1 csubg 14615 . 2  class SubGrp
2 vw . . 3  set  w
3 cgrp 14362 . . 3  class  Grp
42cv 1622 . . . . . 6  class  w
5 vs . . . . . . 7  set  s
65cv 1622 . . . . . 6  class  s
7 cress 13149 . . . . . 6  classs
84, 6, 7co 5858 . . . . 5  class  ( ws  s )
98, 3wcel 1684 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 13148 . . . . . 6  class  Base
114, 10cfv 5255 . . . . 5  class  ( Base `  w )
1211cpw 3625 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2547 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4077 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1623 1  wff SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
Colors of variables: wff set class
This definition is referenced by:  issubg  14621
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