# Image of an involutive subbundle is involutive

Let $$f:M\to N$$ be a surjective submersion smooth map between two smooth manifolds $$M$$ and $$N$$. Consider the differential map $$df:TM\to TN$$ between the tangent bundles. Given an involutive subbundle $$A$$ of $$TM$$. Is $$df(A)$$ an involutive subbundle of $$TN$$?

There is no reason why $$df(A)$$ should even be a subbundle. For instance, let $$f:\mathbb{R}^{2}\rightarrow\mathbb{R}:(x,y)\mapsto x$$ be the projection and take $$A=\text{Span}(\partial_{y}+x\partial_{x})\subset T\mathbb{R}^{2}$$. Then $$df(A)$$ is zero at $$0\in\mathbb{R}$$, but one-dimensional at other points $$p\in\mathbb{R}$$.