Definition 1
Tag 0ACY, permanent. Cite it as [Bourbaki, Tag 0ACY].
Lie Groups and Lie Algebras, chapter III, § 4. Passage from Lie algebras to Lie groups
Let G be a Lie group germ and L its Lie algebra. An exponential mapping of G is any analytic mapping defined on an open neighbourhood of 0 in L, with values in G and satisfying the conditions of Theorem 4.
Theorem 4 implies immediately that, for every Lie group germ G, there exists an exponential mapping of G and that two exponential mappings of G coincide on a neighbourhood of 0.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.