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 $i(X)$.

This idea generalizes to higher cardinals as well. For any set $X$, then the existence of a map as you request is equivalent to the existence of an injection of the Hartog number $\aleph(X)$ of $X$, the smallest ordinal not embedable in $X$, to $P(\kappa)$. Ricky's argument again gives the forward implication (just start with any injection of an ordinal below the Hartog number into $X$, using singletons, and then extend recursively up to the Hartog number). Conversely, if you can inject the Hartog number $\aleph(X)$ into $P(X)$, then let $d_x(i)$ be the first set on this list not in $i(X)$. 

<b>Theorem.</b>(ZF) For any set $X$, the property in the question is equivalent to the assertion that that Hartog number $\aleph(X)$ injects into $P(X)$.