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Theorems · Theorem · combinatorics

Fin.lt_of_injective_of_notMem

∀ {m n : ℕ} (f : Fin m → Fin n), Function.Injective f → ∀ {b : Fin n}, b ∉ Set.range f → m < n

If we have an injective map from Fin m to Fin n whose image does not contain everything, then m < n. See also Fintype.card_lt_of_injective_of_notMem for the generalisation to arbitrary finite types.

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Mathlib.Data.Fin.Pigeonhole
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Foundations
Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound
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Set · cited by 53352SetSet.range · cited by 4705Set.rangeFintype.card_fin · cited by 270Fintype.card_finFintype.card_lt_of_injective_of_notMem · cited by 4Fintype.card_lt_of_inject…Fin.lt_of_injective_of_notMemCITED BYCITES

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