Regular value theorem

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This article gives the statement and possibly proof of a theorem that discusses regular values, critical values, regular points or critical points of a smooth map between differential manifolds

Statement

Let M, N be differential manifolds and p \in N be a regular value of a differentiable map f: M \to N. Then f^{-1}(p) is a submanifold of M.

A slightly stronger version of this result states the following: if there is an open neighbourhood U of p in N such that the rank of the Jacobian is constant for all points in f^{-1}(U), then f^{-1}(p) is a submanifold of M.