ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-iress Unicode version

Definition df-iress 12440
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, defining a function using the base set and applying that, or explicitly truncating the slot before use.

(Credit for this operator, as well as the 2023 modification for iset.mm, goes to Mario Carneiro.)

(Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 7-Oct-2023.)

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

Detailed syntax breakdown of Definition df-iress
StepHypRef Expression
1 cress 12433 . 2  classs
2 vw . . 3  setvar  w
3 vx . . 3  setvar  x
4 cvv 2737 . . 3  class  _V
52cv 1352 . . . 4  class  w
6 cnx 12429 . . . . . 6  class  ndx
7 cbs 12432 . . . . . 6  class  Base
86, 7cfv 5211 . . . . 5  class  ( Base `  ndx )
93cv 1352 . . . . . 6  class  x
105, 7cfv 5211 . . . . . 6  class  ( Base `  w )
119, 10cin 3128 . . . . 5  class  ( x  i^i  ( Base `  w
) )
128, 11cop 3594 . . . 4  class  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>.
13 csts 12430 . . . 4  class sSet
145, 12, 13co 5868 . . 3  class  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w
) ) >. )
152, 3, 4, 4, 14cmpo 5870 . 2  class  ( w  e.  _V ,  x  e.  _V  |->  ( w sSet  <. (
Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) )
161, 15wceq 1353 1  wffs  =  ( w  e.  _V ,  x  e. 
_V  |->  ( w sSet  <. (
Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) )
Colors of variables: wff set class
This definition is referenced by:  reldmress  12492  ressvalsets  12493
  Copyright terms: Public domain W3C validator