The definition of a compound Poisson process and its characteristic function I have are the following:
>Let $\lambda>0$ and $N\sim\text{Poisson}(\lambda T)$. Also, $\{X_i\}_{i=1}^N$ are i.i.d. and independent of $N$. And $\{U_i\}_{i=1}^N$ are i.i.d., $U_i\sim\text{Uniform}([0,T])$, and independent from $X_i,N$. Define:
$$
Y_t\equiv\sum_{i=1}^N\mathbb{1}_{\{U_i\leq t\}}X_i, 0\leq t\leq T
$$
Then $Y_t$ is a compound Poisson process with intensity parameter $\lambda$ and jump pdf $f(x)$.

>The characteristic function of $Y_1$ is:
$$
\mathbb{E}(e^{iuY_1})=e^{\lambda\int(e^{ix}-1)f(x)dx}
$$

Note that the characteristic function I quoted above is for $Y_1$, not $Y_t$. I am trying to show the equality above. I currently have:

$$
\begin{align}
\mathbb{E}(e^{iuY_1})&=\sum_nP(N=n)\mathbb{E}(e^{iuY_1}\mid N=n)\\
&=\sum_nP(N=n)\prod_{j=1}^n\mathbb{E}(e^{iu\mathbb{1}_{\{U_j\leq 1\}}X_j})\quad\text{(by independence)}\\
&=\sum_n P(N=n)\prod_{j=1}\int e^{iux}f(x)dx\quad\text{(by uniform)}
\end{align}
$$

I am not sure how to proceed. Any tips?
Thanks for helping! :D