Extension property for equi-Lebesgue families of functions

Let X be a topological space and (Y, d) be a complete separable metric space. For a family F of functions from X to Y we say that F is equi-Lebesgue if for every ε > 0 there is a cover (F) of X consisting of closed sets such that diam f (F) ≤ ε for all n ∈ N and f ∈ F. We prove that if X is a perfectly normal space, Y is a complete separable metric space and E ⊆ X is an arbitrary set, then every equi-continuous family F ⊆ Y can be extended to an equi-Lebesgue family G ⊆ Y.