Proposition 9
Tag 0B3N, permanent. Cite it as [Bourbaki, Tag 0B3N].
General Topology, chapter I, § 5. Open mappings and closed mappings
Let X, X' be two topological spaces. A necessary and sufficient condition for a mapping f : X → X' to be continuous and closed is that f(Ā) = f(A) for every subset A of X.
The condition is sufficient, for it obviously implies that f is closed, and it also implies that f is continuous by reason of § 2, no. 1, Theorem 1. Conversely, if f is continuous and closed, we have f(A) ⊂ f(Ā) ⊂ f(A) by § 2, no. 1, Theorem 1; also f(Ā) is closed in X' by hypothesis; hence f(Ā) = f(A).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.