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Definition df-subg 13747
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 13766), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 13761), contains the neutral element of the group (see subg0 13757) and contains the inverses for all of its elements (see subginvcl 13760). (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 13744 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 13573 . . 3  class  Grp
42cv 1394 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1394 . . . . . 6  class  s
7 cress 13073 . . . . . 6  classs
84, 6, 7co 6013 . . . . 5  class  ( ws  s )
98, 3wcel 2200 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 13072 . . . . . 6  class  Base
114, 10cfv 5324 . . . . 5  class  ( Base `  w )
1211cpw 3650 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2512 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4148 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1395 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  13750  subgex  13753
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