Sard's Theorem and Generic Transversality

Let \(M,N,P\) be smooth manifolds without boundary and \(F:M\times P\to N\) a smooth map which is transverse to a submanifold \(Q\subset N\). Prove the Transversality Theorem: for almost every \(p\in P\) the map \(F_p:M\to N\) defined by \(F_p(x)=F(x,p)\) is transverse to \(Q\). If you’d like, you can consider the special case when \(M,Q\) are submanifolds of \(P=N=\mathbb {R}^n\) and \(F(x,p)=x+p\).