Problem from Tao's Analysis I.
Exercise 3.4.6. Prove Lemma 3.4.10. (Hint: start with the set $\{0,1\}^X$ and apply the replacement axiom, replacing each function $f$ with the object $f^{-1}(\{1\})$.)
I have read some answers on stackexchange ((exercise from Tao's analysis book) Proof of a lemma relating to power set of X , Finding a correspondence between $\{0,1\}^A$ and $\mathcal P(A)$) but my proof is a little different and I want to know if it's correct and complete, using the provided axioms.
Lemma 3.4.10 Let $X$ be a set. Then the set $\{Y: Y \textit{ is a subset of } X \}$ is a set.
Axiom 3.7 (Replacement). Let $A$ be a set. For any object $x\in A$, and any object $y$, suppose we have a statement $P(x,y)$ pertaining to $x$ and $y$, such that for each $x \in A$ there at most one $y$ for which $P(x,y)$ is true. Then there exists a set $\{y:P(x,y)\textit{ is true for some }x\in A\}$, such that for any object $z$, $$z\in\{y:P(x,y)\textit{ is true for some }x\in A\}\\ \iff > P(x,z) \textit{ is true for some } x\in A$$
Axiom 3.11 (Power set axiom). Let $X$ and $Y$ be sets. Then there exists a set, denoted $Y^X$, which consits of all the functions from $X$ to $Y$, thus $$f\in Y^X \iff \textit{ ($f$ is a function with domain $X$ and range $Y$).}$$
proof. By Axiom 3.11, $\{0,1\}^X$ is a set, namely, $$\{0,1\}^X=\{f: \textit{ $f$ is a function from $X$ to $\{0,1\}$}\}.$$ Then by Axiom 3.7, $$\{A:A=f^{-1}(\{1\} \textit{ for some } f\in \{0,1\}\}\text{ is a set.}$$ We will show that for all sets $A$, $A\in P$ if and only if $A\subseteq X$.
Let $A\subseteq X$, and let $I_A:X\rightarrow \{0,1\}$ be the function such that $$I_A(a)=\begin{cases} 0 \text{, if } a\notin A \\ 1 \text{, if } a\in A \end{cases}.$$ Since $I_A$ is a function from $X$ to $\{0,1\}$, $I_A\in \{0,1\}^X$. We will show that $A=I_A^{-1}(\{1\})$. Suppose $a\in A$. Then $I_A(a)=1$, hence $a\in I_A^{-1}(\{1\})$. Now suppose $a\in I_A^{-1}(\{1\})$. Then $I_A(a)\in\{1\}$, hence $I_A(a)=1$, so $a\in A$.
Thus $A\in P$ if and only if $A\subseteq X$, thus $P$ is the set of all subsets of $X$.
