Definition df-ress 12424

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 goes to Mario Carneiro.)

(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 12417	. 2 class ↾s
2		vw	. . 3 setvar w
3		vx	. . 3 setvar x
4		cvv 2730	. . 3 class _V
5	2	cv 1347	. . . . . 6 class w
6		cbs 12416	. . . . . 6 class Base
7	5, 6	cfv 5198	. . . . 5 class ( Base ` w )
8	3	cv 1347	. . . . 5 class x
9	7, 8	wss 3121	. . . 4 wff ( Base ` w ) C_ x
10		cnx 12413	. . . . . . 7 class ndx
11	10, 6	cfv 5198	. . . . . 6 class ( Base ` ndx )
12	8, 7	cin 3120	. . . . . 6 class ( x i^i ( Base ` w ) )
13	11, 12	cop 3586	. . . . 5 class <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >.
14		csts 12414	. . . . 5 class sSet
15	5, 13, 14	co 5853	. . . 4 class ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. )
16	9, 5, 15	cif 3526	. . 3 class if ( ( Base ` w ) C_ x , w , ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )
17	2, 3, 4, 4, 16	cmpo 5855	. 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 1348	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  12476  ressid2  12477  ressval2  12478