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 $i(X)$.

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

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 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 $\kappa^+\to P(\kappa)$. Ricky's argument again gives the forward direction, and conversely, if there is an injection $\kappa^+\to P(\kappa)$, then we can let $d_X(i)$ be the first set on the list not in $i(X)$.

Perhaps it is true that for any set $X$, the property on $X$ is equivalent to the assertion that the Hartog number $\aleph(X)$, the first ordinal not embedding in $X$, injects into $P(X)$. The converse direction is the same as above, but it isn't clear to me whether one can push Ricky's argument through for this.

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 $\kappa^+\to P(\kappa)$.

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 $i(X)$.

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