I'm struggling to complete a proof. In Moment generating function of $X+Y$ using convolution of $X$ and $Y$, the answer gives
$$m_{X+Y}(t) = \int_{-\infty}^\infty e^{ts} f_{X+Y}(s) \mathop{ds}= \int_{-\infty}^\infty e^{ts} \int_{-\infty}^\infty f_X(s-y) f_Y(y) \mathop{dy} \mathop{ds}, s = x + y$$
Where $x, y$ are independent. The Jacobian from $(s, y)$ to $(x, y)$ I got $\begin{bmatrix}1 &1 \\ 0 &1\end{bmatrix}$ whose determinant is $1$. I think this means
$$m_{X+Y}(t) = \int_{-\infty}^\infty e^{tx}e^{ty} \int_{-\infty}^\infty f_X(x) f_Y(y) \mathop{dy} \mathop{dx}$$
So while I can get this to
$$m_{X+Y}(t) = \int_{-\infty}^\infty e^{tx} f_X(x) \mathop{dx} \cdot \; e^{ty} \int_{-\infty}^\infty f_Y(y) \mathop{dy}$$
I need help getting the $e^{ty}$ inside the integral to complete the proof.