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Definition df-ress 13155
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 13198 for the altered base set, and resslem 13201 (subrg0 15552, ressplusg 13250, subrg1 15555, ressmulr 13261) 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 13149 . 2  classs
2 vw . . 3  set  w
3 vx . . 3  set  x
4 cvv 2788 . . 3  class  _V
52cv 1622 . . . . . 6  class  w
6 cbs 13148 . . . . . 6  class  Base
75, 6cfv 5255 . . . . 5  class  ( Base `  w )
83cv 1622 . . . . 5  class  x
97, 8wss 3152 . . . 4  wff  ( Base `  w )  C_  x
10 cnx 13145 . . . . . . 7  class  ndx
1110, 6cfv 5255 . . . . . 6  class  ( Base `  ndx )
128, 7cin 3151 . . . . . 6  class  ( x  i^i  ( Base `  w
) )
1311, 12cop 3643 . . . . 5  class  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>.
14 csts 13146 . . . . 5  class sSet
155, 13, 14co 5858 . . . 4  class  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w
) ) >. )
169, 5, 15cif 3565 . . 3  class  if ( ( Base `  w
)  C_  x ,  w ,  ( w sSet  <.
( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) )
172, 3, 4, 4, 16cmpt2 5860 . 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 1623 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  13194  ressval  13195
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