Let $X$ & $Y$ be random vectors. Let $Z$=$f(X,Y)$ be a random variable.
Then if $X$ & $Y$ are independent, the following is a well known result:
$P(f(X,Y)\in B\mid Y=y)$ = $P(f(X,y)\in B)$
I need some help to 'formalize' what appears to be an intuitive generalization when $X$ & $Y$ are not assumed to be independent, namely that
$P(f(X,Y)\in B\mid Y=y)$ = $P(f(X,y)\in B\mid Y=y)$
without assuming $Y$ is discrete.
One 'immediate' problem for example is that, I am not sure that the RHS of the above equation is well defined.
For in Breiman's text 'Probability', by Definition 4.7, the conditional probability $P(C\mid Y=y)$ is defined as a measurable function in $y$ satisfying
$P(C, Y\in A )$=$\int_{A} P( C\mid Y=y) \, P_{Y}(dy)$ where $C$ is a measurable subset of the sample space and $A$ is an arbitrary Borel subset in the state space of $Y$.
Notice that $C$ is held fixed in the definition, as $y$ is allowed to vary.
Identifying $C$ with$[f(X,Y)\in B]$ in the LHS of our conjecture, allows us to make sense of $P(f(X,Y)\in B\mid Y=y)$.
But the subset $[f(X,y)\in B]$ changes with changes in $y$, hence at least by the above definition $P(f(X,y)\in B\mid Y=y)$ is not well defined. As an aside, for similar reasons, $E(f(X,y)\in B\mid Y=y)$ is also problematic.
Using the concept of Regular Conditional Probability, I think the statement above can be formalized as follows.
Let $X$ & $Y$ be random vectors. Let $Z$=$f(X,Y)$ be a random variable, and $P_{Y}$ be the marginal distribution of $Y$. Let $Q_{X\mid Y}(\cdot\mid y)$ be a regular conditional distribution for $X$ given $Y=y$.
Then $P(f(X,Y)\in B\mid Y=y)$ = $P(f(X,y)\in B\mid Y=y)$, is I believe an intuitive interpretation of the following 'unproven' statement
$Q_{Z\mid Y}( B\mid y)$= $Q_{X\mid Y}(B_y \mid y)$
where $B$ is any arbitrary Borel subset of $R$, and $B_y$ = $[x \mid f(x,y) \in B]$
and $Q_{Z\mid Y}(\cdot\mid y)$ the regular conditional distribution for $Z=f(X,Y)$ given $Y=y$.
Written alternatively we need to prove that
$P(Z\in B, Y\in A )$=$\int_{A} Q_{Z\mid Y}( B\mid y) \, P_{Y}(dy)$=$\int_{A} Q_{X\mid Y}( B_y\mid y) \, P_{Y}(dy)$
A related problem is to prove the following:
Let $P_{(X,Y)}$ be the joint distribution of $(X,Y)$, $Q_{X\mid Y}(\cdot\mid y)$ be a regular conditional distribution for $X$ given $Y=y$, $P_{Y}$ be the marginal distribution of $Y$, and let $D$ be a given Borel subset in the state space of $(X,Y)$.
Then $P_{(X,Y)}(D )$=$\int Q_{X\mid Y}( D_y\mid y) \, P_{Y}(dy)$ where
$D_y$ = $[x \mid (x,y) \in D]$