I would like to ask about notation, because I think that I am missing something.
It is from book; "The Joy of Sets" - Keith Devlin. There is presented an alternative notation; $ x = \{a | P(a) \} $ which means that $ x $ is a set of all those $ a $ for which $ P(a) $ holds. Does it mean that $ P(a) $ has to evaluate to true?
Then we have definition of intersection of $ x $ which looks like that $ \bigcap x = \{ a | \forall y (y \in x \rightarrow a \in y ) \} $. Let's assume that $ x = \{ \{ 1, 2 \}, \{ 1, 3 \} \} $ then $ \bigcap x $ should be equal to $ \{ 1 \} $ but I can also said that it is equal $ \{ 8 \} $, because $ y = \{ 8 \}$ is not an element of $ x $ so the $ P(a) $ is evaluated to true, so $ \bigcap x = \{ 8 \} $ which is wrong. Can you explain me what I am wrongly mixing? I know that normally I would assume that $ y \in x $ is true, but in this case the $ P(a) $ should be true, shouldn't it?
Yes, when we write $x=\{a\mid P(a)\}$, we mean that $x$ is the set of all objects $a$ such that $P(a)$ is a true statement.
You have $\bigcap x$ defined to be $\{a\mid\forall y\,(y\in x\to a\in y)\}$: $\bigcap x$ is the set of all objects $a$ for which it is true that $a\in y$ whenever $y\in x$. In other words, the definition say precisely that $\bigcap x$ is the set of all objects that are elements of every element of $x$. When $$x=\{\{1,2\},\{1,3\}\}\;,$$ this means that $\bigcap x=\{1\}$: it is true that $\forall y\,(y\in x\to 1\in y)$, and there is no other object $a$ of which we can truly say that $\forall y\,(y\in x\to a\in y)$. In particular, $\forall y\,(y\in x\to 8\in y)$ is not true, because $\{1,2\}\in x$, but $8\notin\{1,2\}$.