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Here is the set-up. Suppose that $\left \{ A_\alpha : \alpha \in \Lambda \right \}$ is a collection of subsets of a set X and $\left \{ B_\beta : \beta \in \Psi \right \}$ is a collection of subsets of a set Y. If $f: X\rightarrow Y$ is a function, then

$$f(f^{-1}(B_\beta)\cap A_\alpha) = B_\beta \cap f(A_\alpha)\;.$$

I'm working to prove that one is a subset of the other to show equality. So far I have proved the forward direction with relative ease but the reverse direction is giving me problems.

My only idea is starting:

Let $x \in B_\beta \cap f(A_\alpha)$, then $x \in B_\beta$ and $x \in f(A_\alpha)$. From here I'm not really sure how to introduce the inverse image of $B_\beta$. Any help would be much appreciated.

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I deleted the general-topology tag, because there’s no actual topology in the question. –  Brian M. Scott Sep 3 '12 at 23:04
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1 Answer

You have $x\in f[A_\alpha]$. This means that there is some $y\in A_\alpha$ such that $x=f(y)$. You also have $x\in B_\beta$, so $f(y)\in B_\beta$, and hence $y\in f^{-1}[B_\beta]$. In other words, $y\in f^{-1}[B_\beta]\cap A_\alpha$. So where is $x=f(y)$?

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Thanks a bunch. –  Mitch Benning Sep 3 '12 at 23:29
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