Definition df-scaf 13789

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 13787	. 2 class .sf
2		vg	. . 3 setvar g
3		cvv 2760	. . 3 class _V
4		vx	. . . 4 setvar x
5		vy	. . . 4 setvar y
6	2	cv 1363	. . . . . 6 class g
7		csca 12701	. . . . . 6 class Scalar
8	6, 7	cfv 5255	. . . . 5 class (Scalar ` g )
9		cbs 12621	. . . . 5 class Base
10	8, 9	cfv 5255	. . . 4 class ( Base ` (Scalar ` g ) )
11	6, 9	cfv 5255	. . . 4 class ( Base ` g )
12	4	cv 1363	. . . . 5 class x
13	5	cv 1363	. . . . 5 class y
14		cvsca 12702	. . . . . 6 class .s
15	6, 14	cfv 5255	. . . . 5 class ( .s ` g )
16	12, 13, 15	co 5919	. . . 4 class ( x ( .s ` g ) y )
17	4, 5, 10, 11, 16	cmpo 5921	. . 3 class ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) )
18	2, 3, 17	cmpt 4091	. 2 class ( g e. _V |-> ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) ) )
19	1, 18	wceq 1364	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  13805