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Definition df-ress 13149
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 13192 for the altered base set, and resslem 13195 (subrg0 15546, ressplusg 13244, subrg1 15549, ressmulr 13255) 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 13143 . 2  classs
2 vw . . 3  set  w
3 vx . . 3  set  x
4 cvv 2789 . . 3  class  _V
52cv 1623 . . . . . 6  class  w
6 cbs 13142 . . . . . 6  class  Base
75, 6cfv 5221 . . . . 5  class  ( Base `  w )
83cv 1623 . . . . 5  class  x
97, 8wss 3153 . . . 4  wff  ( Base `  w )  C_  x
10 cnx 13139 . . . . . . 7  class  ndx
1110, 6cfv 5221 . . . . . 6  class  ( Base `  ndx )
128, 7cin 3152 . . . . . 6  class  ( x  i^i  ( Base `  w
) )
1311, 12cop 3644 . . . . 5  class  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>.
14 csts 13140 . . . . 5  class sSet
155, 13, 14co 5819 . . . 4  class  ( w sSet  <. ( Base `  ndx ) ,  ( x  i^i  ( Base `  w
) ) >. )
169, 5, 15cif 3566 . . 3  class  if ( ( Base `  w
)  C_  x ,  w ,  ( w sSet  <.
( Base `  ndx ) ,  ( x  i^i  ( Base `  w ) )
>. ) )
172, 3, 4, 4, 16cmpt2 5821 . 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 1624 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  13188  ressval  13189
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