Let $X, Y$ be metric spaces with distance functions denoted by $d_X, d_Y$ respectively. Consider a map $f \colon X \rightarrow Y$. I am interested in the following property: for every $x,y,z \in X$, if $d_X(x,y) \leq d_X(x,z)$ then $d_Y(f(x), f(y)) \leq d_Y(f(x), f(z))$. My question is: is this property known and studied? If yes, could you elaborate on the terminology and give references for it? How is it related to $f$ being an isometry?

Thank you in advance for your help.