The cardinality of the union of sets.

Question

The finite union of finite sets is finite. The countable union of countable sets is countable. More generally, is true that if $I$ is an index set of cardinality $\mathfrak{c}$, and $I$ and $X_i$ have same cardinality for every $i$, then the set: $$X=\bigcup_{i \in I}X_i. $$ Has cardinality $\mathfrak{c}$?