If $A\subset Y$,then $f^{-1}(A)=X-f^{-1}(Y-A)?$

Question

If $(x_{\lambda})$ is an ultranet in X and $f:X\rightarrow Y$ then $(f(x_{\lambda}))$ is an ultranet in Y.

Proof: If $A\subset Y$,then $f^{-1}(A)=X-f^{-1}(Y-A),$so ($x_{\lambda}$) is eventually in either $f^{-1}(A)$ or $f^{-1}(Y-A),$ from which it follows that $(f(x_{\lambda}))$ is eventually in either A or $Y-A.$Thus $(f(x_{\lambda}))$ is an ultranet.

The above proof is given in Willard's topolgy.I'm not getting the how to prove

"If $A\subset Y$,then $f^{-1}(A)=X-f^{-1}(Y-A)$"

Answer 1

$f^{-1}(A) = X - f^{-1}(Y-A)$ Proof:

$X = f^{-1}(Y) = f^{-1}(A \cup A^{c}) = f^{-1}(A) \cup f^{-1}(A^{c})$ (disjoint union)

So $(f^{-1}(A^{c}))^{c} = f^{-1}(A)$

And $X - f^{-1}(Y - A) = X - f^{-1}(A^{c}) = (f^{-1}(A^{c}))^{c} = f^{-1}(A)$

Answer 2

$x \in f^{-1}(A)$ iff $f(x) \in A$ iff $\lnot (f(x) \in Y - A)$ iff $x \notin f^{-1}(Y-A)$ iff $x \in X - f^{-1}(Y-A)$.