simp only [Set.inter_singleton_nonempty, Set.image_singleton, Set.inter_self, ne_eq]
case h

⊢ (fun x => 0) '' ({1} ∩ {2}) ≠ (fun x => 0) '' {1} ∩ (fun x => 0) '' {2}
case h

⊢ ¬(fun x => 0) '' ({1} ∩ {2}) = {0}