Definition df-scaf 14735

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 14733	. 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 13493	. . . . . 6 class Scalar
8	6, 7	cfv 5380	. . . . 5 class (Scalar ` g )
9		cbs 13410	. . . . 5 class Base
10	8, 9	cfv 5380	. . . 4 class ( Base ` (Scalar ` g ) )
11	6, 9	cfv 5380	. . . 4 class ( Base ` g )
12	4	cv 1401	. . . . 5 class x
13	5	cv 1401	. . . . 5 class y
14		cvsca 13494	. . . . . 6 class .s
15	6, 14	cfv 5380	. . . . 5 class ( .s ` g )
16	12, 13, 15	co 6088	. . . 4 class ( x ( .s ` g ) y )
17	4, 5, 10, 11, 16	cmpo 6090	. . 3 class ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) )
18	2, 3, 17	cmpt 4193	. 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  14752