How to remove top margin above bigcup

Question

I want to remove the margin above \bigcup symbol :

\begin{displaymath}
\prod_{i\in I} X_i \bydef \left\{\left. x\in\left( \bigcup_{i\in I} X_i \right)^I \right\vert  \: \forall i\in I, x(i)\in X_i \right\}
\end{displaymath}

I don't know how to proceed ? Thank for help :)

Answer 1

I wouldn't autosize any of the "fence" symbols. Instead, I'd use \Big and \bigg sizing directives. Observe that TeX automatically (and correctly) inserts some whitespace around \biggm\vert; this doesn't happen if one writes \middle\vert (or \right\vert).

\documentclass{article}
\begin{document}
\[
\prod_{i\in I} X_i = \biggl\{ x\in\Bigl(\bigcup_{i\in I} X_i \Bigr)^{\!I} 
    \biggm\vert  \forall i\in I, x(i)\in X_i \biggr\}
\]
\end{document} 

Answer 2

is this better?

considering David Carlisle comment you can rewrite your equation into:

\[
\prod_{i\in I} X_i = \left\{ x\in\biggl(\ \bigcup_{i\in I} X_i \biggr)^I \middle\vert  \: \forall i\in I, x(i)\in X_i \right\}
\]