Homotopy fiber of a bundle

Question

Let $f\colon X\to Y$ be a locally trivial bundle with fiber $F$ where $X,Y$ are compact CW-complexes (or even smooth manifolds).

Is it true that $F$ is homotopically equivalent the homotopy fiber of $f$?

A reference would be helpful.

Sorry if this question is not of the research level.

Answer 1

This question was basically already asked on Stackexchange, see this for an answer.

Answer 2

A fiber bundle is a Serre fibration (this is a standard fact, for example, Hatcher, prop. 4.48). Therefore, its fiber over any point is its homotopy fiber.