Definition df-iress 12472

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

Step	Hyp	Ref	Expression
1		cress 12465	. 2 class ↾s
2		vw	. . 3 setvar w
3		vx	. . 3 setvar x
4		cvv 2739	. . 3 class _V
5	2	cv 1352	. . . 4 class w
6		cnx 12461	. . . . . 6 class ndx
7		cbs 12464	. . . . . 6 class Base
8	6, 7	cfv 5218	. . . . 5 class ( Base ` ndx )
9	3	cv 1352	. . . . . 6 class x
10	5, 7	cfv 5218	. . . . . 6 class ( Base ` w )
11	9, 10	cin 3130	. . . . 5 class ( x i^i ( Base ` w ) )
12	8, 11	cop 3597	. . . 4 class <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >.
13		csts 12462	. . . 4 class sSet
14	5, 12, 13	co 5877	. . 3 class ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. )
15	2, 3, 4, 4, 14	cmpo 5879	. 2 class ( w e. _V , x e. _V |-> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )
16	1, 15	wceq 1353	1 wff ↾s = ( w e. _V , x e. _V |-> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )

This definition is referenced by:  reldmress  12525  ressvalsets  12526