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Definition df-sets 13430
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 13431 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 15604, 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 13422 . 2  class sSet
2 vs . . 3  set  s
3 ve . . 3  set  e
4 cvv 2916 . . 3  class  _V
52cv 1648 . . . . 5  class  s
63cv 1648 . . . . . . . 8  class  e
76csn 3774 . . . . . . 7  class  { e }
87cdm 4837 . . . . . 6  class  dom  {
e }
94, 8cdif 3277 . . . . 5  class  ( _V 
\  dom  { e } )
105, 9cres 4839 . . . 4  class  ( s  |`  ( _V  \  dom  { e } ) )
1110, 7cun 3278 . . 3  class  ( ( s  |`  ( _V  \  dom  { e } ) )  u.  {
e } )
122, 3, 4, 4, 11cmpt2 6042 . 2  class  ( s  e.  _V ,  e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u.  {
e } ) )
131, 12wceq 1649 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  13446  setsvalg  13447
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