Does $P_{z} \operatorname{curl} F = P_z \operatorname{curl} P_{xy}F$?

Question

Let $F: \mathbb R^3 \to \mathbb R^3$ and let $P_{xy}F$ be the projection of $F$ onto the $xy$ plane and $P_zF$ the projection of $F$ onto the $z$ axis. Is it true that $$P_{z} \operatorname{curl} F = P_z \operatorname{curl} P_{xy}F$$

Background: I'm interested in $P_z \operatorname{curl} F$, and the problem becomes easier if I instead take $P_z \operatorname{curl} F P_{xy} F$, which should be trivially equal by the definition of curl.

My question is to:

1. Confirm if this is correct?
2. Is there better notation or terminology? How do I say "Since we're only interested in the $z$ component of the curl of $F$, we can pretend that $F$'s $z$ component is zero everywhere?"