partialEquivLimit_comp_inclusion — Mathlib

English

If M is countable, then the set of FG-equivalences from M to M is countable.

Русский

Если множество M счетно, то множество FG-эквивалентностей M к M счетно.

LaTeX

$$$[Countable\ M] \Rightarrow Countable (L.FGEquiv\ M\ M).$$$

Lean4

@[simp]
theorem partialEquivLimit_comp_inclusion {i : ι} :
    (partialEquivLimit S).toEquiv.toEmbedding.comp (Substructure.inclusion (le_iSup _ i)) =
      (Substructure.inclusion (le_iSup _ i)).comp (S i).toEquiv.toEmbedding :=
  by
  simp only [partialEquivLimit, Equiv.comp_toEmbedding, Embedding.comp_assoc]
  rw [Equiv_isup_symm_inclusion]
  congr

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