Theorem staffn

Description: The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015)

	Ref	Expression
Hypotheses	staffval.b	|- B = ( Base ` R )
	staffval.i	|- .* = ( *r ` R )
	staffval.f	|- .xb = ( *rf ` R )
Assertion	staffn	|- ( .* Fn B -> .xb = .* )

Proof

Step	Hyp	Ref	Expression
1		staffval.b	|- B = ( Base ` R )
2		staffval.i	|- .* = ( *r ` R )
3		staffval.f	|- .xb = ( *rf ` R )
4	1 2 3	staffval	|- .xb = ( x e. B |-> ( .* ` x ) )
5		dffn5	|- ( .* Fn B <-> .* = ( x e. B |-> ( .* ` x ) ) )
6	5	biimpi	|- ( .* Fn B -> .* = ( x e. B |-> ( .* ` x ) ) )
7	4 6	eqtr4id	|- ( .* Fn B -> .xb = .* )