Composition Proof I am trying to study functions in math and learning some basic proofs. In numerous places I have seen this: $$(f \circ\ g) ^{-1}(u) = g^{-1}(f^{-1}(u))$$
I know this is true as well, having used it in numerous places in middle and high school. Is there any way of proving this definition though using logical steps? Thank you!

Assume functions $g:A\mapsto B, f:B\mapsto C$ are bijective (therefore invertible).
By definition : $(f\circ g)(x)=f(g(x))$ 
Applying inversion:
$$g^{-1}(f^{-1}((f\circ g)(x)))=x$$
Let : $x=(f\circ g)^{-1}(u)$ which means also $u=(f\circ g)(x)$
Therefore …$$g^{-1}(f^{-1}(u))=(f\circ g)^{-1}(u)$$
