Answer
Assume C $\subseteq$ D and assume x $\in$ $\ f^-1$(C)
by definition of range f(y) = x , x $\in$ C
(by definition of intersection x $\in$ C and x $\in$ D)
hence, f(y) = x , x $\in$ D
which implies x $\in$ $\ f^-1$(D)
in conclusion x $\in$ $\ f^-1$(C) implies x $\in$ $\ f^-1$(D)
hence by definition of intersection $\ f^-1$(C) $\subseteq$ $\ f^-1$(D)
Work Step by Step
Assume C $\subseteq$ D and assume x $\in$ $\ f^-1$(C)
by definition of range f(y) = x , x $\in$ C
(by definition of intersection x $\in$ C and x $\in$ D)
hence, f(y) = x , x $\in$ D
which implies x $\in$ $\ f^-1$(D)
in conclusion x $\in$ $\ f^-1$(C) implies x $\in$ $\ f^-1$(D)
hence by definition of intersection $\ f^-1$(C) $\subseteq$ $\ f^-1$(D)
