I am reading *Fluctuations of Levy Processes with Applications* by A.E. Kyprianou and I am having struggles understanding a part in the proof of theorem 5.6. Let $Y$ be a subordinator and $\mathbf{e}$ an independent exponential random variable with parameter $\eta$. Let $X$ be the killed subordinator associated with $Y$ and $\mathbf{e}$, i.e., the stochastic process defined by
$$
X_t =
\begin{cases}
Y_{t}, & \text{if } t<\mathbf{e}\\
\partial & \text{if } \mathbf{e}\geq t,
\end{cases}
$$
where $\partial$ is a cemetery point. Let $\tau_x^+$ be the first-passage time at $x$, that is,
$$
\tau_x^+ = \inf\{t>0: X_t>x\}.
$$
In the proof of the aforementioned theorem, the following equality is stated without any explanation:
$$
\mathbb{E}\left[f(X_{\tau_x^+}-x)g(x-X_{\tau_x^+-})\right]=\mathbb{E}\left[\int_{[0,\infty)}\int_{(0,\infty)}e^{-\eta t}\phi(t,\theta)N(dt\times d\theta)\right]
$$
where $f$ and $g$ are two continuous functions vanishing at infinite with $f(0)=g(0)=0$, $N$ is the Poisson random measure associated with $Y$ and $\phi$ is defined by
$$
\phi(t,\theta)= 1_{(Y_{t-}\leq x)}1_{(Y_{t-}+\theta> x)}f(Y_{t-}+\theta- x)g(x-Y_{t-}).
$$
I would like to ask if anyone would have a bit of insight into why this is true.

I have found the proof of the same theorem in other resources (for example, in Bertoin's *Levy Processes*), but they all use the same equality without any argumentation so as to why it is true. I can see why the equality is true in the case when $Y$ is a compound Poisson process with drift, and I feel like I could get the result by a limiting argument, but I feel like there should be a more direct way of seeing this.

Thank you for any kind of help you might provide.