Cantor's diagonalization construction, on a certain view, furnishes functions $$d_X:{\rm Injections}(X,P(X))\rightarrow P(X)$$ that satisfy $\forall X\forall i\ \ d_X(i)\not\in i(X)$

In ZF, can one prove the existence of such functions with the added requirement that $d_x(i)$ actually depends only on the image $i(X)$?