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Definition df-ress 13466
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 13509 for the altered base set, and resslem 13512 (subrg0 15865, ressplusg 13561, subrg1 15868, ressmulr 13572) 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 13460 . 2  classs
2 vw . . 3  set  w
3 vx . . 3  set  x
4 cvv 2948 . . 3  class  _V
52cv 1651 . . . . . 6  class  w
6 cbs 13459 . . . . . 6  class  Base
75, 6cfv 5446 . . . . 5  class  ( Base `  w )
83cv 1651 . . . . 5  class  x
97, 8wss 3312 . . . 4  wff  ( Base `  w )  C_  x
10 cnx 13456 . . . . . . 7  class  ndx
1110, 6cfv 5446 . . . . . 6  class  ( Base `  ndx )
128, 7cin 3311 . . . . . 6  class  ( x  i^i  ( Base `  w
) )
1311, 12cop 3809 . . . . 5  class  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>.
14 csts 13457 . . . . 5  class sSet
155, 13, 14co 6073 . . . 4  class  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w
) ) >. )
169, 5, 15cif 3731 . . 3  class  if ( ( Base `  w
)  C_  x ,  w ,  ( w sSet  <.
( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) )
172, 3, 4, 4, 16cmpt2 6075 . 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 1652 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  13505  ressval  13506
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