Conditional transformation of variables

I've seen a trick for finding the p.d.f of $$r(X,Y)$$ where $$X$$ and $$Y$$ are r.v's by first calculating the cdf i.e $$P(r(X,Y) \leq l)$$ and then differentiating to find the pdf. So if $$\Omega = \{(x,y) | r(x,y) \leq l)$$ then $$P(r(X,Y) \leq l) = \int_{\Omega} f_{X,Y}(x,y).$$

However what if i want the pdf of $$r(X,Y)$$ conditioned on $$X=t$$ ? I'm thinking i might do something like this,

First calculate $$f_{Y|X}(y|x=t)$$ to get the conditional pdf. Then let $$\Omega' = \{y | r(t,y) \leq l\}$$ then the cdf we are interested in is $$\int_{\Omega'} f_{Y|X}(y|x=t)$$, this gives me the conditional cdf of $$r(t,y)$$, i can then differentiate to find the pdf.

Is this correct?