I have been reading about the rate distortion function in which the fundamental limit of compression of a random variable $X$ to another random variable $Y$ taking values on smaller alphabet within distortion $D$ is given as $$R(D)=\inf_{P(y|x), ~ \text{s.t.} Ed(x,y)\leq D } I(X;Y)$$ where $d: \mathcal{X}\times \mathcal{Y}\to \mathbb{R}_{+}$ is a distortion measure. I am wondering if there is any way to replace the condition of $Ed(x,y)\leq D$ with another condition of sort $D(P_X||P_Y)< \gamma$ where $D$ is a divergence measure between two probability distributions of $X$ and $Y$ (for example think of $f$-divergence).
Basically, I am asking about a following "if and only if" relation: $$Ed(x,y)\leq D \Leftrightarrow D(P_X||P_Y)< \gamma.$$
Any help is appreciated.