Reducing to regular cardinals in c.c.c. implies same cardinals and cofinalities

Question

In Chapter 14 of Jech's Set Theory, he asserts the following:

Definition 14.33. A forcing notion $P$ satisfies the countable chain condition (c.c.c.) if every antichain in $P$ is at most countable.

Theorem 14.34. If $P$ satisfies the countable chain condition then $V$ and $V[G]$ have the same cardinals and cofinalities.

The proof begins with

It suffices to show that if $\kappa$ is a regular cardinal then $\kappa$ remains regular in $V[G]$.

The proof continues, but my main point of confusion lies at the very beginning. Why is it sufficient to prove that regular cardinals remain regular? How does it follow from this that all cardinals and cofinalities are preserved?

Answer 1

If $\mu$ has cofinality $\kappa<\mu$, and we changed the cofinality of $\mu$, then we had to have changed the cofinality of $\kappa$ as well, since the new cofinal sequence will define a cofinal sequence in $\kappa$.

This means that $\kappa$ is no longer a regular cardinal, as it was in the ground model, being the cofinality of $\mu$. And quite often, it might be that $\kappa$ is not a cardinal at all in $V[G]$.

So if a forcing preserves regularity, then it preserves cofinalities as well.