Cardinality of the set of functions defined on a finite set .

If $F$ be the set of all functions defined on $I_n =\{ 1 , 2 , 3 ,..., n\} , n\in \mathbb{N} $ with range $B\subseteq I^+$( set of positive integers ) . Then

(a) $F$ is countable

(b) $F$ is uncountable .

(c) $F$ is infinite .

(d) $F$ is countable if $B$ is finite .

My attempt :- Let $Y\subseteq B$

If $Range(f)=Y$, then$|Y|=|f(I_n)|\le |I_n| =n $

So for any $Y \subseteq B, s.t |Y| \le n$

Let $F(Y)=\{ f\in F | Range(f)=Y\} $

Then $|F(Y)|$ is finite and $F=\cup\{ F(Y) | Y \subseteq B ,|Y| \le n \} $ .

As the set of all finite subsets of $B$ is countable and countable union of finite sets is countable , so $F$ is countable.

I request you to cross check my proof and suggest any better alternatives. Thanks .