On Page 65, Set Theory, Jech(2006), Collection Principle is formulated as follows:
$\forall{X}\exists{Y}(\forall{u}\in{X})[\exists{v} \psi(u, v, p) \to(\exists{v\in{Y}}) \psi(u, v, p)]$
($p$ is a parameter)
Even with help of some explanations below it, I still have difficulty understanding it.
Here's how far I understand it.
$X$ is an arbitrary set of index. $u$ is a generic element of $X$. $Y$ is a set whose element is determined by $X$. If there exists $v$ that satisfy $ \psi(u, v, p)$, then $v$ belongs to $Y$.
My interpretation deviates from the textbook, which is as follows:
Let $C_u=\{v:\psi(u, v, p)\}$, if $C_u \neq \emptyset$, then $C_u \cap Y \neq \emptyset$.
Another question of mine is if so, is it correct to replace $v$ in $(\exists{v\in{Y}}) \psi(u, v, p)]$ with other arbitrary variable, say, $w$? Then the Collection Principle should be modifies as: $\forall{X}\exists{Y}(\forall{u}\in{X})[\exists{v} \psi(u, v, p) \to(\exists{w\in{Y}}) \psi(u, w, p)]$