Quotient map & adjuction space

Suppose f : X→Y is a quotient map between spaces such that it maps a subset A→X injectively into Y and maps X\A onto Y\p(A). Then Y is homeomorphic to the adjuction space X∪_p|X\A(Y\p(A)).

pf> Let A'=X\A, B=p(A), B'=Y\p(A). Let π:X∪B'→X∪_p|A'B' be the quotient map defining the adjunction space, and let φ:X∪_p|A'B'→Y be a map defined in the obvious way. Then clearly φ is bijective. I'll show it's homeomorphism.
Suppose U⊆Y is open. π^-1φ^-1(U)=p^-1(U) so is open in X, and π^-1φ^-1(U)∩B' =U∩B' is obviously open in B' so π^-1φ^-1(U) is open in X∪B', and hence φ^-1(U) is open in X∪_p|A'B'.
Now suppose V⊆Y is a subset such that φ^-1(V) is open. Then π^-1φ^-1(V) is open. Since p^-1(V)=π^-1φ^-1(V), V is open in Y. QED