Vrftsjtryk

Assume we have a set of sets $textbf{P}={ textbf{P}_j }_{j=1}^{n}$ where $textbf{P}_j={P_j^i }_{i=1}^{m_j}$ for every $textbf{P}_j in textbf{P}$. Now, we want to build a set of $P_j^i$ for every $(i,j) in j times m_j$ for every $jin n$ from $textbf{P}$. In plain words, we are flattening an unaligned nested set, or an unaligned matrix.

Here is expression [1]:
$${pintextbf{P}_j|textbf{P}_j in textbf{P} } $$
where the condition is actually on the member's domain.

And another expression [2]:
$$bigcup_{j=1}^n { p|p in textbf{P}_j }, where textbf{P}_j in textbf{P}$$

Which expression makes more sense and why? Are there any other expressions for this case?

The notationally simplest way to describe the desired set is as
$$
bigcup textbf{P}.
$$
The only downside is that some mathematicians, in my experience, aren't very comfortable with the general set theoretic union and prefer to union sets over a given index set.

If you want to take that route, notice that
$$
bigcup_{j=1}^n { p mid p in textbf{P}_j } = bigcup_{j=1}^n textbf{P}_j,
$$
so you can simplify your 2nd expression.

Finally, your first expression strikes me as rather inelegant and borderline confusing. You could rewrite it as
$$
{p mid exists x in textbf{P} colon p in x }
$$
but that's really just an overly complicated way to write $bigcup textbf{P}$.