Definition df-ress 15865

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 15930 for the altered base set, and resslem 15933 (subrg0 18787, ressplusg 15993, subrg1 18790, ressmulr 16006) 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

Step	Hyp	Ref	Expression
1		cress 15858	. 2 class ↾s
2		vw	. . 3 setvar w
3		vx	. . 3 setvar x
4		cvv 3200	. . 3 class _V
5	2	cv 1482	. . . . . 6 class w
6		cbs 15857	. . . . . 6 class Base
7	5, 6	cfv 5888	. . . . 5 class ( Base ` w )
8	3	cv 1482	. . . . 5 class x
9	7, 8	wss 3574	. . . 4 wff ( Base ` w ) C_ x
10		cnx 15854	. . . . . . 7 class ndx
11	10, 6	cfv 5888	. . . . . 6 class ( Base ` ndx )
12	8, 7	cin 3573	. . . . . 6 class ( x i^i ( Base ` w ) )
13	11, 12	cop 4183	. . . . 5 class <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >.
14		csts 15855	. . . . 5 class sSet
15	5, 13, 14	co 6650	. . . 4 class ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. )
16	9, 5, 15	cif 4086	. . 3 class if ( ( Base ` w ) C_ x , w , ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )
17	2, 3, 4, 4, 16	cmpt2 6652	. 2 class ( w e. _V , x e. _V |-> if ( ( Base ` w ) C_ x , w , ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) ) )
18	1, 17	wceq 1483	1 wff ↾s = ( w e. _V , x e. _V |-> if ( ( Base ` w ) C_ x , w , ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) ) )

This definition is referenced by:  reldmress  15926  ressval  15927