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Definition df-sets 13149
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 13150 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 15321, 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 13141 . 2  class sSet
2 vs . . 3  set  s
3 ve . . 3  set  e
4 cvv 2790 . . 3  class  _V
52cv 1623 . . . . 5  class  s
63cv 1623 . . . . . . . 8  class  e
76csn 3642 . . . . . . 7  class  { e }
87cdm 4689 . . . . . 6  class  dom  { 
e }
94, 8cdif 3151 . . . . 5  class  ( _V 
\  dom  {  e } )
105, 9cres 4691 . . . 4  class  ( s  |`  ( _V  \  dom  {  e } ) )
1110, 7cun 3152 . . 3  class  ( ( s  |`  ( _V  \  dom  {  e } ) )  u.  {
e } )
122, 3, 4, 4, 11cmpt2 5822 . 2  class  ( s  e.  _V ,  e  e.  _V  |->  ( ( s  |`  ( _V  \  dom  {  e } ) )  u.  {
e } ) )
131, 12wceq 1624 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  13165  setsvalg  13166
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