Definition df-ress 11956

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 11949	. 2 class ↾s
2		vw	. . 3 setvar w
3		vx	. . 3 setvar x
4		cvv 2681	. . 3 class _V
5	2	cv 1330	. . . . . 6 class w
6		cbs 11948	. . . . . 6 class Base
7	5, 6	cfv 5118	. . . . 5 class ( Base ` w )
8	3	cv 1330	. . . . 5 class x
9	7, 8	wss 3066	. . . 4 wff ( Base ` w ) C_ x
10		cnx 11945	. . . . . . 7 class ndx
11	10, 6	cfv 5118	. . . . . 6 class ( Base ` ndx )
12	8, 7	cin 3065	. . . . . 6 class ( x i^i ( Base ` w ) )
13	11, 12	cop 3525	. . . . 5 class <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >.
14		csts 11946	. . . . 5 class sSet
15	5, 13, 14	co 5767	. . . 4 class ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. )
16	9, 5, 15	cif 3469	. . 3 class if ( ( Base ` w ) C_ x , w , ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )
17	2, 3, 4, 4, 16	cmpo 5769	. 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 1331	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  12006  ressid2  12007  ressval2  12008