Reference for Turing to many-one reductions I am looking for a reference on `reducing' Turing reductions to many-one reductions.  I have in mind a statement of the following form (similar enough statements would also satisfy me):

Theorem.  If $\mathsf{A}\leq_T^f \mathsf{B}$, then $\mathsf A\leq_m^{2^f}\mathsf B^{tt}$.

where "$\leq_T^f$" and "$\leq_m^f$" denote respectively Turing and many-one reductions in deterministic time $f(n)$, and "$\mathsf{B}^{tt}$" denotes a `truth table' variant of the language $\mathsf{B}$, which evaluates a Boolean combination of checks "$x\in\mathsf{B}$".

Proof idea for the statement:  Simulate the $f(n)$-time bounded oracle Turing machine used in the Turing reduction: it's easy enough to obtain a nondeterministic Turing transducer also in time $f(n)$ that guesses the answers of the $\mathsf B$ oracle and writes a conjunction of checks "$x\in\mathsf B$" or "$x\not\in\mathsf{B}$" on the output, to be evaluated by an $\mathsf B^{tt}$ machine.  This transducer can be determinized by exploring both outcomes of the oracle calls, and handling them through disjunctions in the output; it now works in time $2^{f(n)}$.

Oddly enough, I cannot seem to find any related result in complexity textbooks.

Edit: renamed "$A\mathsf B$" into "$\mathsf B^{tt}$" to emphasize the relation with truth tables, as pointed out by @MarkusBläser.
I'm pretty sure you can find this in the book Bounded Queries in Recursion Theory by Martin and Gasarch, but I don't have access to a copy now to check. 

(Your statement gives an upper bound of $2^f$; most of that book is about lower bounding such quantities, so the upper bound must be in there somewhere, probably very near the beginning.)

PS - I agree with M. Blaser's comment: you're basically talking about truth-table reductions without using that terminology.