I came across some papers 1, 2 and others which use the following formula
$ p(y) = \int_{\mathcal{X}} \, d^{n}x \, \delta(F(\vec{x}) - y) \, p(\vec{x}) $
where $p(y)$ is the PDF of the variable defined by $y=F(\vec{x})$ , also $p(\vec{x})$ is the PDF of the variables $(x_1,x_2,...)$, $\mathcal{X}$ is the $n$-dimensional volume of the variables $\vec{x}$, and $\delta$ is the dirac delta function (see equation 3 in ref 1 if you prefer a non-vector notation).
However there were no proofs in any of the papers. My question is, are there any sources proving this formula? Or is there some quick intuitive proof?
My attempt at a proof was as follows. From Theorem 6.1.5 of Hormander's book, "The Analysis of Linear Partial Differential Operators I", I set $\rho = y-F(\vec{x})$ as the $n+1$-dimensional function in the theorem. From there, after computing the euclidean surface measure, I believe one can show that the cumulative distribution obeys:
$ P(a) = \int_{-\infty}^{a} \,\,p(y)\,dy = \int_{\mathcal{X}'} \,d^{n}x\,\, p(\vec{x}) $
where $\mathcal{X}' := \{x \in \mathcal{X} | F(x) < a \}$ is a suitably restricted volume. $P(a)$ is the required CDF for the function of random variable $y$.
My formal analysis is a bit rusty so it's possible I got something wrong. Also, I was finding this method of proof not very intuitive. My guess is there's a simple trick I'm missing, potentially something obvious from measure theory.