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 $i(X)$? 



Cantor's diagonal argument and ZF
If there is such a function then there is an injection from $\omega\_1$ to $2^{\omega}$.
  

(Set $\: X = \omega \:$, $\:$ send the finite ordinals to the corresponding singletons,
  

then extend to $\omega\_1$ with [transfinite recursion](https://planetmath.org/TransfiniteRecursion).)


If there is such a function and the continuum hypothesis holds then $\: 2^{\aleph\_0} = \aleph\_1 \:$.


It is [consistent with ZF that the continuum hypothesis holds and](https://math.stackexchange.com/questions/43861/relationship-between-continuum-hypothesis-and-special-aleph-hypothesis-under-zf/43877#43877) $\: 2^{\aleph\_0} \neq \aleph\_1 \:$.


Therefore ZF does not prove the existence of such a function.


