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Definition df-ress 13252
Description: Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the  Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range (like  Ring), defining a function using the base set and applying that (like  TopGrp), or explicitly truncating the slot before use (like  MetSp).

(Credit for this operator goes to Mario Carneiro).

See ressbas 13295 for the altered base set, and resslem 13298 (subrg0 15651, ressplusg 13347, subrg1 15654, ressmulr 13358) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Assertion
Ref Expression
df-ress  |-s  =  ( w  e.  _V ,  x  e. 
_V  |->  if ( (
Base `  w )  C_  x ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) ) )
Distinct variable group:    x, w

Detailed syntax breakdown of Definition df-ress
StepHypRef Expression
1 cress 13246 . 2  classs
2 vw . . 3  set  w
3 vx . . 3  set  x
4 cvv 2864 . . 3  class  _V
52cv 1641 . . . . . 6  class  w
6 cbs 13245 . . . . . 6  class  Base
75, 6cfv 5337 . . . . 5  class  ( Base `  w )
83cv 1641 . . . . 5  class  x
97, 8wss 3228 . . . 4  wff  ( Base `  w )  C_  x
10 cnx 13242 . . . . . . 7  class  ndx
1110, 6cfv 5337 . . . . . 6  class  ( Base `  ndx )
128, 7cin 3227 . . . . . 6  class  ( x  i^i  ( Base `  w
) )
1311, 12cop 3719 . . . . 5  class  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>.
14 csts 13243 . . . . 5  class sSet
155, 13, 14co 5945 . . . 4  class  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w
) ) >. )
169, 5, 15cif 3641 . . 3  class  if ( ( Base `  w
)  C_  x ,  w ,  ( w sSet  <.
( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) )
172, 3, 4, 4, 16cmpt2 5947 . 2  class  ( w  e.  _V ,  x  e.  _V  |->  if ( (
Base `  w )  C_  x ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) ) )
181, 17wceq 1642 1  wffs  =  ( w  e.  _V ,  x  e. 
_V  |->  if ( (
Base `  w )  C_  x ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) ) )
Colors of variables: wff set class
This definition is referenced by:  reldmress  13291  ressval  13292
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