When defining $\Sigma_i^P$ or $\Pi_i^P$ completeness, we want to use a reduction that fulfills the following property: If $L' \leq_p L$ and $L \in \Sigma_i^P$ or $\Pi_i^P$ respectively, then $L'$ is also $\Sigma_i^P$ or $\Pi_i^P$.
I can see how Karp-reductions fulfill this requirement for the complexity class $P$. How could one proof that Karp-reduction fulfill this requirement for all other complexity classes in the polynomial hierachy?
Suppose that $f$ is the polynomial reduction between $L'$ and $L$, i.e. $x \in L' \Leftrightarrow f(x) \in L$. If $L \in \Sigma_k^p$ then $y\in L \Leftrightarrow \exists z_1 \forall z_2 \ldots M(y, z_1, \ldots z_k) = 1$. Then $$x\in L \Leftrightarrow f(x) \in L \Leftrightarrow \exists z_1 \forall z_2 \ldots M(f(x), z_1, \ldots z_k) = 1 \Leftrightarrow \exists z_1 \forall z_2 \ldots P(x, z_1, \ldots z_k) = 1$$ $P$ first computes $f$, then applies $M$, so it's polynomial.
