Renaming_Mapping_is_Bijection

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let f: S → T be a mapping. The renaming mapping r: S/R_f → Im(f), defined as r([x]_{R_f}) = f(x), is a bijection onto the image of f.

Why this node was retired

The posted statement is

theorem Renaming_Mapping_is_Bijection {α β : Type _} (f : α → β) : Function.Bijective (Quotient.lift f (fun a b h => h) : Quotient (Setoid.ker f) → β) := by sorry

The quotient lift has codomain all β, rather than the image of f.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For f:α→β, the induced map Quotient(Setoid.ker f)→Set.range f is a bijection. If retaining codomain β, require f surjective.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.