map_reflexive — Mathlib

English

If r is transitive and f respects r-steps, then Map r f f is transitive.

Русский

Если r транзитивно и f сохраняет шаги r, то Map r f f транзитивно.

LaTeX

$$$\forall r:\alpha\to\alpha\to \mathrm{Prop},\; \mathrm{Transitive}(r) \Rightarrow \forall f:\alpha\to\beta,\; (\forall x,y, f x = f y \Rightarrow r x y) \Rightarrow \mathrm{Transitive}(\mathrm{Map}(r,f,f))$$$

Lean4

theorem map_reflexive {r : α → α → Prop} (hr : Reflexive r) {f : α → β} (hf : f.Surjective) :
    Reflexive (Relation.Map r f f) := by
  intro x
  obtain ⟨y, rfl⟩ := hf x
  exact ⟨y, y, hr y, rfl, rfl⟩

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