Assume that $X$ is a Poisson process on $S\subseteq\mathbb{R}^{d}$ with intensity function $\rho$. Then: $$\tag{1} \mathbb{P}(X\cap B\in F\mid N(B)=n)=\int_{B}\dots\int_{B}\mathbb{1}[\{x_{1},\dots,x_{n}\}\in F]\prod_{i=1}^{n}\frac{\rho(x_{i})}{\mu(B)}\,\mathrm{d}x_{1}\dots\mathrm{d}x_{n}. $$
The definition of a Poisson point process that I'm working with is as follows: A point process $X$ on $S$ is a Poisson point process with intensity function $\rho$ if: (a) $N(B)\sim\mathrm{po}(\mu(B))$, and (b) given $N(B)=n$ for $n\in\mathbb{N}$, the $n$ points in $X\cap B$ form a binomial point process on $B$ with pdf $u\mapsto\rho(u)/\mu(B)$ with $u\in B$.
Note that I've intentioanlly avoided addressing what exactly $F$ and $B$ are to spare some mathematical technicalities that I believe are irrelevant for this question.
My question: How exactly do I see that (1) is true? I'm well aware that it is a consequence of (b), however, I can't seem to work out the exact details.