Definition df-scaf 14294

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 14292	. 2 class .sf
2		vg	. . 3 setvar g
3		cvv 2800	. . 3 class _V
4		vx	. . . 4 setvar x
5		vy	. . . 4 setvar y
6	2	cv 1394	. . . . . 6 class g
7		csca 13153	. . . . . 6 class Scalar
8	6, 7	cfv 5324	. . . . 5 class (Scalar ` g )
9		cbs 13072	. . . . 5 class Base
10	8, 9	cfv 5324	. . . 4 class ( Base ` (Scalar ` g ) )
11	6, 9	cfv 5324	. . . 4 class ( Base ` g )
12	4	cv 1394	. . . . 5 class x
13	5	cv 1394	. . . . 5 class y
14		cvsca 13154	. . . . . 6 class .s
15	6, 14	cfv 5324	. . . . 5 class ( .s ` g )
16	12, 13, 15	co 6013	. . . 4 class ( x ( .s ` g ) y )
17	4, 5, 10, 11, 16	cmpo 6015	. . 3 class ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) )
18	2, 3, 17	cmpt 4148	. 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  14310