I expect this should be a basic property of regular conditional densities/stochastic kernels, but somehow I am having trouble verifying this.
Suppose we have random variables $X$ and $Y$ with (smooth) joint probability density $p(x,y)$ and (smooth) conditional density for X given Y, namely $p(x|y)$. Then, is it true that: $$p(x|y \in A) = \frac{1}{\mathbb{P}(Y \in A)} \int_A p(x|y) p(y) dy $$
I would write something like
$$p(x \mid y \in A)=\dfrac{\int\limits_{y \in A} p(x,y) \,dy}{\int\limits_{z}\int\limits_{y \in A} p(z,y) \,dy\, dz}$$ providing that the denominator is positive, where $z$ ranges over all possible values of $x$
The numerator is pretty much your $\int\limits_A p(x\mid y) \,p(y) \,dy $ and the denominator your $\mathbb{P}(Y \in A)$, so your expression works too.