The image of a smooth embedding is an embedded submanifold

Fix $p\in M$. By [F1] the map $F$ is an immersion, so [L1] gives charts near $p$ and $F(p)$ in which $F$ becomes $u\mapsto (u,0)$.

In those target coordinates, the image of a small neighbourhood of $p$ is exactly the coordinate slice ${\mathbb{R}}^m\times \{0\}$. Because [F1] also says that $F$ is a homeomorphism onto its image, we may shrink the neighbourhood in the target so that no other points of $M$ map into that same slice patch. Hence the image is locally a slice chart in the sense of [F2].

Since every point of $F(M)$ has such a slice neighbourhood, $F(M)$ is an embedded $m$-submanifold. The corestriction $F:M\to F(M)$ is already a homeomorphism by [F1], and in the local coordinates of step 1.1 it is the identity on the ${\mathbb{R}}^m$ factor, so it is a diffeomorphism.