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Definition df-sets 13154
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 13155 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 15326, 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 13146 . 2  class sSet
2 vs . . 3  set  s
3 ve . . 3  set  e
4 cvv 2788 . . 3  class  _V
52cv 1622 . . . . 5  class  s
63cv 1622 . . . . . . . 8  class  e
76csn 3640 . . . . . . 7  class  { e }
87cdm 4689 . . . . . 6  class  dom  {
e }
94, 8cdif 3149 . . . . 5  class  ( _V 
\  dom  { e } )
105, 9cres 4691 . . . 4  class  ( s  |`  ( _V  \  dom  { e } ) )
1110, 7cun 3150 . . 3  class  ( ( s  |`  ( _V  \  dom  { e } ) )  u.  {
e } )
122, 3, 4, 4, 11cmpt2 5860 . 2  class  ( s  e.  _V ,  e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  { e } ) )  u.  {
e } ) )
131, 12wceq 1623 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  13170  setsvalg  13171
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