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Definition df-sets 13245
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 13246 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 15419, 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 13237 . 2  class sSet
2 vs . . 3  set  s
3 ve . . 3  set  e
4 cvv 2864 . . 3  class  _V
52cv 1641 . . . . 5  class  s
63cv 1641 . . . . . . . 8  class  e
76csn 3716 . . . . . . 7  class  { e }
87cdm 4768 . . . . . 6  class  dom  {
e }
94, 8cdif 3225 . . . . 5  class  ( _V 
\  dom  { e } )
105, 9cres 4770 . . . 4  class  ( s  |`  ( _V  \  dom  { e } ) )
1110, 7cun 3226 . . 3  class  ( ( s  |`  ( _V  \  dom  { e } ) )  u.  {
e } )
122, 3, 4, 4, 11cmpt2 5944 . 2  class  ( s  e.  _V ,  e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u.  {
e } ) )
131, 12wceq 1642 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  13261  setsvalg  13262
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