Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)

Question

Where can I find a rigorous proof of the Dupire formula (for example, using using Gyöngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not find one. To save space, if such a reference of a rigorous derivation for the Dupire formula exists, posting the reference should suffice.