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Definition df-subg 14972
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14990), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14985), contains the neutral element of the group (see subg0 14981) and contains the inverses for all of its elements (see subginvcl 14984). (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 14969 . 2  class SubGrp
2 vw . . 3  set  w
3 cgrp 14716 . . 3  class  Grp
42cv 1652 . . . . . 6  class  w
5 vs . . . . . . 7  set  s
65cv 1652 . . . . . 6  class  s
7 cress 13501 . . . . . 6  classs
84, 6, 7co 6110 . . . . 5  class  ( ws  s )
98, 3wcel 1727 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 13500 . . . . . 6  class  Base
114, 10cfv 5483 . . . . 5  class  ( Base `  w )
1211cpw 3823 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2715 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4291 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1653 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  14975
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