If $A$ is an empty set, how should I understand $\forall x\in A$? It might look quite stupid, but I had become little confused when understanding empty functions. Anyway, my question is,
If there is a statement $P(x)$ starting with "for $\forall x\in A$,..." and $A$ is an empty set, should I understand this as because the assumption is false, the conclusion is absolutely true? If not, how should I?

Well, the place where I got stuck was this:
For every set X, there exists a unique empty function $f : \emptyset \rightarrow X$. To prove this I should set two empty functions $f_1, f_2$, and show that $\forall x\in \emptyset$, $f_1(x)=f_2(x)$. When thinking as I stated above, since the assumption is false, the conclusion is true. But instead if we think about a statement $\forall x\in \emptyset$, $f_1(x)\neq f_2(x)$, this may be also true....(?)

Imagine this:
Everytime I have played the lottery I won the jackpot!
Why is this true? Am I the luckiest person on the planet? No. I just have never played the lottery.
That is: the set of all times T that I played the lottery is empty ... which is exactly why the claim  $\forall t \in T: Jackpot!(t)$ is true.
And yes, unfortunately it is also true that $\forall t \in T: \neg Jackpot!(t)$
