I'm trying to prove that if $X$ and $Y$ are first countable topological spaces and $Z$ is another topological space. Then if $\hat{f}:X \to C_c(Y,Z)$ is continuous we also have that $f:X\times Y \to Z$ is continuous. Where $C_c(Y,Z)$ denotes the set of continuous functions from $Y$ to $Z$ equipped with the compact-open topology, and $\hat{f}:X\to C_c(Y,Z)$ is defined by $\hat{f}(x)(y)=f(x,y)$ for each $x \in X$ and $y \in Y$.
What I have so far is that if $U$ is an open set in $Z$ and $(x_0,y_0) \in f^{-1}[U]$ then $f(x_0,y_0) \in U$. So that $\hat{f}(x_0)(y_0) \in U$ and $\hat{f}(x_0) \in M(\{y_0\},U)$, where $M(\{y_0\},U)$ denotes the set of continuous functions that send the elements of $\{y_0\}$ to $U$. Then I tried to use that $X$ is first countable saying that there exists an open set $B_n$ in $X$ such that $\hat{f}(x_0) \in \hat{f}[B_n]\subseteq M(\{y_0\},U)$. But then I became stuck since this $B_n$ has some dependence on $y_0$. Any ideas on how to continue? Am I heading in the right direction?