Definition df-scaf 14167

Description: Define the functionalization of the .s operator. This restricts the value of .s to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.)

Assertion

Ref	Expression
df-scaf	|- .sf = ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )

Distinct variable group:   x, g, y

Detailed syntax breakdown of Definition df-scaf

Step	Hyp	Ref	Expression
1		cscaf 14165	. 2 class .sf
2		vg	. . 3 setvar g
3		cvv 2776	. . 3 class _V
4		vx	. . . 4 setvar x
5		vy	. . . 4 setvar y
6	2	cv 1372	. . . . . 6 class g
7		csca 13027	. . . . . 6 class Scalar
8	6, 7	cfv 5290	. . . . 5 class (Scalar ` g )
9		cbs 12947	. . . . 5 class Base
10	8, 9	cfv 5290	. . . 4 class ( Base ` (Scalar ` g ) )
11	6, 9	cfv 5290	. . . 4 class ( Base ` g )
12	4	cv 1372	. . . . 5 class x
13	5	cv 1372	. . . . 5 class y
14		cvsca 13028	. . . . . 6 class .s
15	6, 14	cfv 5290	. . . . 5 class ( .s ` g )
16	12, 13, 15	co 5967	. . . 4 class ( x ( .s ` g ) y )
17	4, 5, 10, 11, 16	cmpo 5969	. . 3 class ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) )
18	2, 3, 17	cmpt 4121	. 2 class ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )
19	1, 18	wceq 1373	1 wff .sf = ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )

This definition is referenced by:  scaffvalg  14183