connectedComponentIn — Mathlib

English

If x ∈ F and y ∈ F with y ∈ connectedComponentIn F x, then connectedComponentIn F x = connectedComponentIn F y.

Русский

Если x ∈ F и y ∈ F и y ∈ connectedComponentIn F x, то connectedComponentIn F x = connectedComponentIn F y.

LaTeX

$$$y \in connectedComponentIn F x \Rightarrow connectedComponentIn F x = connectedComponentIn F y$$$

Lean4

theorem connectedComponentIn {x : α} {F : Set α} (h : IsPreconnected F) (hx : x ∈ F) : connectedComponentIn F x = F :=
  (connectedComponentIn_subset F x).antisymm (h.subset_connectedComponentIn hx subset_rfl)

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