Proving Inverse Function of Union Property

[Geekster]

Disclaimer: I might have some problems getting my LaTeX code to work properly so please bear with me while I figure out how to properly use the forum software.

Homework Statement

The exercise is to prove the following statements.

Suppose that f:X \rightarrow Y, the following statement is true.
If \{G_{\alpha} : \alpha \in A\} is an indexed family of subsets of Y, then f^{-1}(\bigcup_{\alpha \in A} G_\alpha) =\bigcup_{\alpha \in A} f^{-1}( G_\alpha).

Homework Equations

The relevant information in this case is the definitions. In this case I need to know what the definition of an inverse function is.

DEF: Suppose f:X \rightarrow Y and A \subset Y. f^{-1}(A) = \{x \in X: f(x) \in A\}

The Attempt at a Solution

The solution I've been looking thus far is a point wise argument.

Choose t \in f^{-1}(\bigcup_{\alpha \in A} G_\alpha). So by definition we know
t \in \{x \in X: f(x) \in (\bigcup_{\alpha \in A} G_\alpha)\}. And here is where I'm kind of stuck. I need to some how get my chosen element to be in the other set, therefore making the one set a subset of the other. Then complete the converse of the argument to finish the proof.

Any ideas on where I should be looking, or what I should be thinking about here?

 

[StatusX]

f(x) \in \cup_{\alpha \in A} G_\alpha is equivalent to "there is some \alpha \in A such that f(x) \in G_\alpha". Also, you don't have to worry about doing the converse if every step you take is an equivalence.