English
If f is injective in both arguments, then the cardinality of image₂ f s t equals the product of the cardinals: |image₂ f s t| = |s|·|t|.
Русский
Если f инъективна по каждому аргументу, то кардинальность image₂ f s t равна произведению кардинальностей: |image₂ f s t| = |s|·|t|.
LaTeX
$$card_image₂ (hf) (s) (t) : #(image₂ f s t) = #s * #t$$
Lean4
theorem card_image₂_iff : #(image₂ f s t) = #s * #t ↔ (s ×ˢ t : Set (α × β)).InjOn fun x => f x.1 x.2 :=
by
rw [← card_product, ← coe_product]
exact card_image_iff