I want to show that the Hartogs operation $\chi(\cdot)$, which sends every set $X$ to a well ordered set $\chi(X)$ not injectable into $X$, is monotone with respect to $\leq_c$. I have come up with a proof idea, but am not sure whether I'm heading in the right direction.
First some definitions. For any set $A$ let \begin{equation*} WO(A) := \{U =(\text{field}~(U), \leq_U): U~ \text{is a well ordered set and field(U)} \subseteq A\} \end{equation*}
Let $\sim_A$ be the restriction of the definite condition of isomorphism to $WO(A)$. Since $\sim_A$ is an equivalence relation we may define $\chi(A) := (h(A), \leq_{\chi(A)})$, where \begin{equation*} h(A) := \{ [U]_{\sim_A}: U \in WO(A)\} \end{equation*} and
\begin{equation*} [U]_{\sim_A} \leq_{\chi(A)} [V]_{\sim_A} \Leftrightarrow U \leq_o V. \end{equation*} Here,$[U]_{\sim_A}$ is the equivalence class of $U$ with respect to $\sim_A$ and $U \leq_o V$ means that there is an isomorphism from $U$ onto an initial segment of $V$, which may be an improper segment.
Now, what I want to prove is that for any sets $A,B$:
\begin{equation*} A \leq_c B \Rightarrow \chi(A) \leq_o \chi(B) \end{equation*}
My proof idea goes as follows. If $\pi$ is an injection from $A$ to $B$ we let $\rho([U]_{\sim_A}) := [\pi[U]]_{\sim_B}$.
The range of $\rho$ seems right since $\pi[U]$ is well ordered by the relation $x \leq_{\pi[U]}y \Leftrightarrow \pi^{-1}(x) \leq_U \pi^{-1}(y)$. However, I have a hard time working out the rest. Therefore my question: Is my idea for proof on the right track? Thanks in advance.