For sets $A$ and $B$, let $f: A’ \rightarrow B’, A’ \subseteq A$ and $B’ \subseteq B,$ be called a partial function. Show that the set of all partial functions from $A$ to $B$ is a set. Use only the power set axiom, axiom of replacement, and union.
Note that this is from Tao’s Analysis text and Cartesian products have not yet been defined.
This question has been asked before here but the answers do not follow Tao’s definition of function equality. Namely two functions must have the same ranges to be considered equal (ie if $Y$ and $Y’$ are the ranges of two functions $f, g$, respectively, the functions cannot be equal even if their inverse images are equal).
Best Answer
Tao proves that if $A$ is a set, then $\{X\mid X\subseteq A\}$ is also a set.
For every fixed $Y\subseteq B$, consider the function $F(X)=Y^X$, and by Replacement, the set $\{Y^X\mid X\subseteq A\}$ exists. For each $Y\subseteq B$.
Next, define the function $G(Y)=\{Y^X\mid X\subseteq A\}$, and again by Replacement the set $\{G(Y)\mid Y\subseteq B\}$ exists.
Finally, apply the Union axiom (two times).