let $f$, $g$ injective functions
$f:E\rightarrow F$ and $g:F\rightarrow E$
consider this set defined as follow
$\mathcal F=\left\{C\subset E, g\left(F-f(C)\right)\subseteq E-C\right\}$
i have proved that $\mathcal F$ is non empty
how i can prove that $K \in \mathcal L$ where $K=\bigcup\limits_{C\in \mathcal F}C$
please help me with this question