Are immersed submanifolds something like local manifolds the same way manifolds are locally Euclidean?

Question

For a (smooth) manifold $M$, the locally Euclidean property goes:

We have that for each $p \in M$, there exists a positive integer $n_p$ such that there exists a neighborhood $U_p$ of $p$ in $M$ such that $U_p$ is diffeomorphic to an open subset $V$ of $\mathbb R^{n_p}$ (If $M$ has a dimension $n$, since in some textbooks, not all manifolds have uniform dimension, then $n=n_p = n_q$ for all $p, q \in M$).

If we view $U_p$ and $V$ as regular/embedded submanifolds with codimension 0 (of their respective spaces $M$ and $\mathbb R^{n_p}$), then I think we have an analogous local-manifold property for an immersed submanifold $P$:

Let $M$ be a manifold such that $P$ is an immersed submanifold of $M$. For each $p \in P$, there exists a subset $W_p$ of $P$ that contains $p$ such that $W_p$ is a regular/an embedded submanifold of $M$ (I'm not too sure about the with or without dimension here for $M$ or $W_p$.).

• Edit: I changed $W_p$ from open to arbitrary subset and then to submanifold (of $M$) and removed ideas of "diffeomorphism" for arbitrary subsets of manifolds.

Question 0: Is this property correct?

Question 1: Assuming the property is correct, is this proof correct? Assuming the property is incorrect, then what's the error in this proof?

1. By definition of $P$ being an immersed submanifold, there exists manifolds $N$ and $M$ and a smooth map $F: N \to M$ such that $F$ is an immersion with image $F(N)=P$. (I'm not sure if $N$ and $M$ necessarily have dimensions.)

2. Let $p \in P = F(N)$. Then there exists $q \in N$ such that $F(q)=p$.

3. For $q$ in (2), with immersions as equivalent to local embeddings, there exists, by (1), a neighborhood $S_q$ of $q$ in $N$ such that $F|_{S_q}: S_q \to M$ is an embedding.

4. The image of $F|_{S_q}$, $F|_{S_q}(S_q)=F(S_q)$, is a regular/an embedded submanifold of $M$, by (3).

5. Choose $W_p = F(S_q)$, by (4).

• Edit: I removed Question 2 asking for a property where $W_p$ is an immersed submanifold of $M$ because $W_p$ is an immersed submanifold since $W_p$ is a regular/an embedded submanifold of $M$.