isOpen_iff — Mathlib

English

If e.IsImage s t, then IsOpen(e.source ∩ s) is equivalent to IsOpen(e.target ∩ t).

Русский

Если e.IsImage s t, то IsOpen(e.source ∩ s) эквивалентно IsOpen(e.target ∩ t).

LaTeX

$$IsImage(e, s, t) \\Rightarrow IsOpen(e.source \\cap s) \\iff IsOpen(e.target \\cap t)$$

Lean4

theorem isOpen_iff (h : e.IsImage s t) : IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t) :=
  ⟨fun hs => h.symm_preimage_eq' ▸ e.symm.isOpen_inter_preimage hs, fun hs =>
    h.preimage_eq' ▸ e.isOpen_inter_preimage hs⟩

Похожие утверждения