Given a random vector $(X,Y) \in AC$ with density $\gamma_{(X,Y)}(x,y) $ we set
$$\gamma_{(X|Y)}(x,y) := \frac{ \gamma_{(X,Y)}(x,y) }{ \gamma_{Y}(y)} \mathbb{1}_{(\gamma_Y(y) > 0)} (y), \qquad (x,y) \in \mathbb{R}^n \times \mathbb{R} $$
And we call
$$\mu_{X|Y}(H,y) := \int_H \gamma_{(X|Y)}(x,y) \,dx, \qquad H \in B_d, y \in R $$
A regular version of the conditional distribution of $X$ given $Y$.
In this theoretical probabilistic context could someone tell me which theorem allows us to claim results on the conditional densities such as that
$$\gamma_{(X+aY|Y)}(x,y) = \gamma_{(X|Y)}(x - ay,y) $$
and that $$\gamma_{(X,Y)}(x,y) = \gamma_{(X-Y, Y)}(x-y,y)$$