Definition df-scaf 14626

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 14624	. 2 class .sf
2		vg	. . 3 setvar g
3		cvv 2821	. . 3 class _V
4		vx	. . . 4 setvar x
5		vy	. . . 4 setvar y
6	2	cv 1401	. . . . . 6 class g
7		csca 13434	. . . . . 6 class Scalar
8	6, 7	cfv 5377	. . . . 5 class (Scalar ` g )
9		cbs 13352	. . . . 5 class Base
10	8, 9	cfv 5377	. . . 4 class ( Base ` (Scalar ` g ) )
11	6, 9	cfv 5377	. . . 4 class ( Base ` g )
12	4	cv 1401	. . . . 5 class x
13	5	cv 1401	. . . . 5 class y
14		cvsca 13435	. . . . . 6 class .s
15	6, 14	cfv 5377	. . . . 5 class ( .s ` g )
16	12, 13, 15	co 6085	. . . 4 class ( x ( .s ` g ) y )
17	4, 5, 10, 11, 16	cmpo 6087	. . 3 class ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) )
18	2, 3, 17	cmpt 4192	. 2 class ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )
19	1, 18	wceq 1402	1 wff .sf = ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )

This definition is used by:  scaffvalg  14643