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Definition df-ress 13481
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 13524 for the altered base set, and resslem 13527 (subrg0 15880, ressplusg 13576, subrg1 15883, ressmulr 13587) 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 13475 . 2  classs
2 vw . . 3  set  w
3 vx . . 3  set  x
4 cvv 2958 . . 3  class  _V
52cv 1652 . . . . . 6  class  w
6 cbs 13474 . . . . . 6  class  Base
75, 6cfv 5457 . . . . 5  class  ( Base `  w )
83cv 1652 . . . . 5  class  x
97, 8wss 3322 . . . 4  wff  ( Base `  w )  C_  x
10 cnx 13471 . . . . . . 7  class  ndx
1110, 6cfv 5457 . . . . . 6  class  ( Base `  ndx )
128, 7cin 3321 . . . . . 6  class  ( x  i^i  ( Base `  w
) )
1311, 12cop 3819 . . . . 5  class  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>.
14 csts 13472 . . . . 5  class sSet
155, 13, 14co 6084 . . . 4  class  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w
) ) >. )
169, 5, 15cif 3741 . . 3  class  if ( ( Base `  w
)  C_  x ,  w ,  ( w sSet  <.
( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) )
172, 3, 4, 4, 16cmpt2 6086 . 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 1653 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  13520  ressval  13521
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