I'm trying to solve exercise 12.1 in Prof. Monk's lectures on set theory which asks me to show that $Fn(\omega_1, 2, \omega_1)$ preserves cardinals larger than $\omega_2$. Now, the only method in the chapter which would seem feasible is to show that the poset satisfies the $\omega_2$-c.c. However, this can't be used, by the following argument: the poset contains $2^{\omega}$, which is a strong antichain of cardinality possibly larger than $\omega_2$. Questions:
1) Is there an error in the argument above?
2) Can the preservation of cardinals be shown by another method or is there an error in the problem?
I'm not necessarily asking for a complete solution. Thanks in advance.