I would like to know why this equality $f_x,_y (x, y; θ) = f_xy (x; θ) × f_y(y; θ)$ holds.
so, why the joint density of two random variables is equal to the conditional density of the first given the second times the second?
It's a given axiom in every book I consulted, but I would like the proof.
This property of probability densities is based on the so-called Law of Total Probability. The proof can be found here: https://proofwiki.org/wiki/Total_Probability_Theorem
The Law of Total Probability can be applied to densities, since densities give the likelihood of an event occurring.