English
A symmetric/anticommutative variant of image₂_union_inter_subset_union, relating unions and intersections under a commutation condition.
Русский
Симметричное/антикоммутативное преобразование image₂_union_inter_subset_union при определённых условиях коммутативности.
LaTeX
$$$\\forall f,s,s',t,t': Finset,\\ (hf):\\ (∀ a b, f a b = f b a) \\Rightarrow image_2 f (s \\cup s') (t \\cap t') \\subseteq image_2 f s t \\cup image_2 f s' t'$$$
Lean4
@[simp (default + 1)] -- otherwise `simp` doesn't use `forall_mem_image₂`
theorem sup'_image₂_le {g : γ → δ} {a : δ} (h : (image₂ f s t).Nonempty) :
sup' (image₂ f s t) h g ≤ a ↔ ∀ x ∈ s, ∀ y ∈ t, g (f x y) ≤ a := by rw [sup'_le_iff, forall_mem_image₂]
