English
If hs is a unique MDiffOn on s and f is MF-differentiable with dense derivative range, then image has unique MDiffOn.
Русский
Если hs — уникальное MDiffOn на s и f имеет MF-дериватив внутри s с плотным диапазоном, то образ имеет уникальное MDiffOn.
LaTeX
$$Same as 119521 with different variation$$
Lean4
/-- If `s` has the unique differential property, `f` is differentiable on `s` and its derivative
at every point of `s` has dense range, then `f '' s` has the unique differential property.
This version uses the `HasMFDerivWithinAt` predicate. -/
theorem image_denseRange' (hs : UniqueMDiffOn I s) {f : M → M'} {f' : M → E →L[𝕜] E'}
(hf : ∀ x ∈ s, HasMFDerivWithinAt I I' f s x (f' x)) (hd : ∀ x ∈ s, DenseRange (f' x)) :
UniqueMDiffOn I' (f '' s) :=
forall_mem_image.2 fun x hx ↦ (hs x hx).image_denseRange (hf x hx) (hd x hx)
