piCongrLeft_preimage_pi — Mathlib

English

A general lemma: for a bijection f, the preimage under piCongrLeft equals the pi over the image by f.

Русский

Общее лемма: для биекции f предобраз под piCongrLeft равен pi по образу через f.

LaTeX

$$$$ f.\\piCogrLeft α \\,-1' (f '' s).\\pi t = s.\\pi (\\lambda i. t (f i)). $$$$

Lean4

theorem piCongrLeft_preimage_pi (f : ι' ≃ ι) (s : Set ι') (t : ∀ i, Set (α i)) :
    f.piCongrLeft α ⁻¹' (f '' s).pi t = s.pi fun i => t (f i) :=
  by
  apply Set.ext
  rw [← (f.piCongrLeft α).symm.forall_congr_right]
  simp

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