Proposition 1.5.30.  Let X, Y be sets, f : X ↔ Y be a bijection, x ∈ X. Then:
f-1(f(x)) = x
Proof.
f-1 ∘ f = idX by definition of f-1 ⇒def ∀ z ∈ X : f-1(f(z)) = z ⇒x ∈ X f-1(f(x)) = x