Suppose we have a 1-Lipschitz function $f$ such that 1-Lipschitzness is preserved, with $D_A(f(X), f(Y)) \leq D_B(X, Y)$ for some metric spaces $A$ and $B$.

Does this also imply that $I(f(X); f(Y)) = I(X;Y)$? It seems intuitive that this would be the case if the 1-lipschitz function is injective, but I don't know if this is true, so that the mutual information is preserved under such invertible transformation. Is there a way to prove this?