English
EquicontinuousOn F S is equivalent to EquicontinuousOn ((↑) : range F → X → α) S.
Русский
ЭконтинуированностьOn через диапазон эквивалентна той же через диапазон.
LaTeX
$$$\\operatorname{EquicontinuousOn} F S \\iff \\operatorname{EquicontinuousOn} \\bigl((\\uparrow) : \\operatorname{range} F \\to X \\to \\alpha\\bigr) S.$$$
Lean4
/-- A family `𝓕 : ι → X → α` is equicontinuous on `S` iff the function `swap 𝓕 : X → ι → α` is
continuous on `S` *when `ι → α` is equipped with the topology of uniform convergence*. This is
very useful for developing the equicontinuity API, but it should not be used directly for other
purposes. -/
theorem equicontinuousOn_iff_continuousOn {F : ι → X → α} {S : Set X} :
EquicontinuousOn F S ↔ ContinuousOn (ofFun ∘ Function.swap F : X → ι →ᵤ α) S := by
simp_rw [EquicontinuousOn, ContinuousOn, equicontinuousWithinAt_iff_continuousWithinAt]
