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Definition df-sets 13477
Description: Set one or more components of a structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 13478 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. Or df-mgp 15651, which takes a ring and overrides its addition operation with the multiplicative operation, so that we can consider the "multiplicative group" using group and monoid theorems, which expect the operation to be in the  +g slot instead of the  .r slot. (Contributed by Mario Carneiro, 1-Dec-2014.)
Assertion
Ref Expression
df-sets  |- sSet  =  ( s  e.  _V , 
e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u. 
{ e } ) )
Distinct variable group:    e, s

Detailed syntax breakdown of Definition df-sets
StepHypRef Expression
1 csts 13469 . 2  class sSet
2 vs . . 3  set  s
3 ve . . 3  set  e
4 cvv 2958 . . 3  class  _V
52cv 1652 . . . . 5  class  s
63cv 1652 . . . . . . . 8  class  e
76csn 3816 . . . . . . 7  class  { e }
87cdm 4880 . . . . . 6  class  dom  {
e }
94, 8cdif 3319 . . . . 5  class  ( _V 
\  dom  { e } )
105, 9cres 4882 . . . 4  class  ( s  |`  ( _V  \  dom  { e } ) )
1110, 7cun 3320 . . 3  class  ( ( s  |`  ( _V  \  dom  { e } ) )  u.  {
e } )
122, 3, 4, 4, 11cmpt2 6085 . 2  class  ( s  e.  _V ,  e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u.  {
e } ) )
131, 12wceq 1653 1  wff sSet  =  ( s  e.  _V , 
e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u. 
{ e } ) )
Colors of variables: wff set class
This definition is referenced by:  reldmsets  13493  setsvalg  13494
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