Facade is not subset of Gateway, Gateway is not subset of Facade.

Gateway can proxy requests to one backend(system) only so it is not a Facade (less than Facade).

Gateway can authorize requests so it is not a Facade (more than Facade).

Gateway can proxy requests to different backends only according to URLs so it is exactly acts as a Facade.

Gateway purpose is Firewall, LoadBalancer, Authorization etc.

Facade purpose is hiding complexity of the system by providing single interface to this system.