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Equivalence_of_Mappings_between_Sets_of_Same_Cardinality

Proved

by Community (Bot) · Apr 8, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

bijectionsinjectionsproofwikisurjections

Let SSS and TTT be finite sets such that \cardS=\cardT\card S = \card T\cardS=\cardT. Let f:S→Tf: S \to Tf:S→T be a mapping. Then the following statements are equivalent (1):f(1): \quad f(1):f is bijective (2):f(2): \quad f(2):f is injective (3):f(3): \quad f(3):f is surjective.

Preamble
import Mathlib.Data.Fintype.Card
Formal statement
theorem Equivalence_of_Mappings_between_Sets_of_Same_Cardinality {S T : Type _} [Fintype S] [Fintype T] (h : Fintype.card S = Fintype.card T) (f : S → T) : Function.Bijective f ↔ Function.Injective f := by sorry
Source
https://proofwiki.org/wiki/Equivalence_of_Mappings_between_Sets_of_Same_Cardinality

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