Let me point out that in the case $X=\mathbb{N}$, the assertion seems to be simply equivalent to the existence of an injection $\omega_1\to \mathbb{R}$. Ricky has cleverly proved the forward implication. But conversely, if there is a such an injection of $\omega_1\to\mathbb{R}$, then we can define $d_X(i)$ to be the first real not in $i(X)$.
I guess this idea generalizes to higher well-ordered cardinals as well. If $X=\kappa$, then the existence of a map as you request is equivalent to the existence of an injection $\kappa^+\to P(\kappa)$.