reference request - Truncated exact sequence of homotopy groups

This is a question about a name of a very useful lemma,
that permits one in particular to show that smooth birational complex projective
varieties have isomorphic fundamental groups.
If this lemma has no name, I would like at least to have a reference (if it exits).
The lemma can be seen as a
truncated version of the basic fact, that if we have a locally trivial
fibration (say of finite dimensional CW complexes) $Fto Eto B$ then
we get a long exact sequence

$to pi_i(F)to pi_i(E)to pi_i(B)to pi_{i-1}(F)to$

Lemma. Let $Eto B$ be a surjective map of finite dimensional $CW$ complexes,
such that every fiber is connected, simply connected and is a
deformation retract of a small neighbourhood.
Then $pi_1(E)=pi_1(B)$.

Question.
Do you know the name of such a lemma, or of some of its generalizations? Is there a reference for this?

The result about $pi_1$ of birationaly equivalent varieties follows
since any birational transformation can be decomposed in blow-ups
and blow downs along smooth submanifolds. And it is not hard
to check that the conditions of lemma are satisfied for such
elementary blow ups.