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Definition df-subg 12961
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2m 12980), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 12975), contains the neutral element of the group (see subg0 12971) and contains the inverses for all of its elements (see subginvcl 12974). (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 12958 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 12809 . . 3  class  Grp
42cv 1352 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1352 . . . . . 6  class  s
7 cress 12455 . . . . . 6  classs
84, 6, 7co 5872 . . . . 5  class  ( ws  s )
98, 3wcel 2148 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 12454 . . . . . 6  class  Base
114, 10cfv 5215 . . . . 5  class  ( Base `  w )
1211cpw 3575 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2459 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4063 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1353 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  12964  subgex  12967
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