A Boolean function takes truth values (0 or 1) as argument, not elements of an arbitrary set X.

A single truth valuation? (You are saying "the" and "valuation" in singular.) What do you mean by identification between a set X and a function (since a truth valuation is a function)?

Do you mean $v_x$ for a concrete $x$ or $v$ as a function of $x$? The function $v_x$ is from $FB(X)$ to {0, 1} and is certainly not one-to-one.

One frequently denotes {0, 1} by 2 and the set of functions from $A$ to $B$ by $B^A$, since the number of elements in $B^A$ is $|B|^{|A|}$, where $|Z|$ denotes the number of elements in some $Z$. Then $FB(X)=2^X$. Are you trying to prove that $v$ is a bijection from $X$ to $2^{2^X}$? This cannot happen because $2^{2^{|X|}}$ is much larger than $|X|$.