Definition df-scaf 13786

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 13784	. 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 12698	. . . . . 6 class Scalar
8	6, 7	cfv 5254	. . . . 5 class (Scalar ` g )
9		cbs 12618	. . . . 5 class Base
10	8, 9	cfv 5254	. . . 4 class ( Base ` (Scalar ` g ) )
11	6, 9	cfv 5254	. . . 4 class ( Base ` g )
12	4	cv 1363	. . . . 5 class x
13	5	cv 1363	. . . . 5 class y
14		cvsca 12699	. . . . . 6 class .s
15	6, 14	cfv 5254	. . . . 5 class ( .s ` g )
16	12, 13, 15	co 5918	. . . 4 class ( x ( .s ` g ) y )
17	4, 5, 10, 11, 16	cmpo 5920	. . 3 class ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) )
18	2, 3, 17	cmpt 4090	. 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  13802