Suppose that a Lie group $G$ acts smoothly on a manifold $X$. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function theorem.
Theorem A. Let $x$ be a point of $X$ and define $F:G \to X$ by $F(g)=gx$. Then there are open neighbourhoods $U \subset G$ of the identity and $V \subset X$ of $x$ such that $F(U)$ is a closed submanifold of $V$.
Questions:
(i) Is it possible to prove Theorem A by using the implicit function theorem (without applying the constant rank theorem)?
(ii) Atiyah-Bott use an infinite-dimensional version of Theorem A in §14: Fix a complex (smooth) vector bundle $E$ over a closed Riemann surface $M$ with a Hermitian metric. Let $G$ be the group $(\mathcal{G}^c)^k$ of complex gauge transformations of class $L^2_k$ and $X$ the space $\mathcal{A}^{k-1}$ of unitary connections of class $L^2_{k-1}$ ($k \geq 2$). According to Atiyah-Bott, we can prove Theorem A in this case by using the implicit function theorem for Banach manifolds. (We may need the Fredholm property of the derivative $dF$.)
How do we prove Theorem A in this case?
(iii) Do we have an infinite-dimensional version of the constant rank theorem (which can be applied to Question (ii))?