This is probably an unusual case for this site. I have to note that I'm not a complete beginner in mathematics. I have seriously studied abstract algebra and analysis. Most of my studies have been successful. Before now.
However, my interests have required of me to learn set theory a little more seriously.
And I encountered that my proof skills are not that perfect, even though they were enough for more "concerete" subjects.
For example, if I want to prove that
For any function $f: X \to Y$ for any family $\{ A_i \}_{i \in I}$ of subsets of $X$ the following is true:
$f( \bigcup_{i \in I} A_i ) = \bigcup_{ i \in I} f( A_i ) $
The proof should go like this:
First, we check for the case $I = \varnothing$. Then, we assume that $I \neq \varnothing$ and go like this:
$y \in f( \bigcup_{i \in I} A_i ) \Leftrightarrow \exists x \in \bigcup_{i \in I} A_i \Leftrightarrow \exists x \in X, ( \exists i \in I, x \in A_i ) : f(x) = y$
To prove the equivalence, I need to prove the following:
$\exists i \in I: (\exists x \in A_i: y = f(x))$
So, how do I deal with similar situations in general? Note that I'm not only asking about how to prove the aformentioned case, I want to be prepared if something else arises involving multiple quatifiers in the future. Is there a particular set of such properties?
Another problem is that I'm not currently interested in mathematical logic. I only want to know the bare minimum needed for proofs in other areas of mathematics. So, please, don't reference me to long books on logic itself.