Definition df-scaf 13846

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 13844	. 2 class .sf
2		vg	. . 3 setvar g
3		cvv 2763	. . 3 class _V
4		vx	. . . 4 setvar x
5		vy	. . . 4 setvar y
6	2	cv 1363	. . . . . 6 class g
7		csca 12758	. . . . . 6 class Scalar
8	6, 7	cfv 5258	. . . . 5 class (Scalar ` g )
9		cbs 12678	. . . . 5 class Base
10	8, 9	cfv 5258	. . . 4 class ( Base ` (Scalar ` g ) )
11	6, 9	cfv 5258	. . . 4 class ( Base ` g )
12	4	cv 1363	. . . . 5 class x
13	5	cv 1363	. . . . 5 class y
14		cvsca 12759	. . . . . 6 class .s
15	6, 14	cfv 5258	. . . . 5 class ( .s ` g )
16	12, 13, 15	co 5922	. . . 4 class ( x ( .s ` g ) y )
17	4, 5, 10, 11, 16	cmpo 5924	. . 3 class ( x e. ( Base ` (Scalar ` g ) ) , y e. ( Base ` g ) |-> ( x ( .s ` g ) y ) )
18	2, 3, 17	cmpt 4094	. 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  13862