Check if $A \subseteq B \land C \subseteq D \Rightarrow A \cap C \subseteq B \cap D$
Consider an arbitrary object $x$. Assume that $x$ is in $A$ and that $x$ is in $C$. Then, by Modus Ponens, we can deduce that $x$ is in $B$ and in $D$. And so the fact that $x$ is in $A$ and $C$ simultaneously entails the fact that this object must be in $B$ and $C$ at the same time as well. The fact that $x$ is in $A$ and $C$, by definition, means that $x \in A\cap C$ and by deduction we got that $x \in B \cap D$
Is my solution correct?