Definition df-scaf 14248

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 14246	. 2 class .sf
2		vg	. . 3 setvar g
3		cvv 2799	. . 3 class _V
4		vx	. . . 4 setvar x
5		vy	. . . 4 setvar y
6	2	cv 1394	. . . . . 6 class g
7		csca 13108	. . . . . 6 class Scalar
8	6, 7	cfv 5317	. . . . 5 class (Scalar ` g )
9		cbs 13027	. . . . 5 class Base
10	8, 9	cfv 5317	. . . 4 class ( Base ` (Scalar ` g ) )
11	6, 9	cfv 5317	. . . 4 class ( Base ` g )
12	4	cv 1394	. . . . 5 class x
13	5	cv 1394	. . . . 5 class y
14		cvsca 13109	. . . . . 6 class .s
15	6, 14	cfv 5317	. . . . 5 class ( .s ` g )
16	12, 13, 15	co 6000	. . . 4 class ( x ( .s ` g ) y )
17	4, 5, 10, 11, 16	cmpo 6002	. . 3 class ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) )
18	2, 3, 17	cmpt 4144	. 2 class ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )
19	1, 18	wceq 1395	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  14264