Claim: Suppose that $f$ is a conformal map from the unit disk to the unit disk, $f(z_1) = w_1$ , $f(z_2) = w_2$, then we it can be derived that $|\frac{z_1-z_2}{1-\bar z_1 z_2}| = |\frac{w_1-w_2}{1-\bar w_1 w_2}|$. Let $z_1$ approaches $z_2$, we have $\frac{|dz|}{1-|z|^2}= \frac{|dw|}{1-|w|^2}$, so can conclude that the Riemannian metric $ds= \frac{2|dz|}{1-|z|^2}$ is invariant under the conformal mapping.
Questions:
What does $dz$ actually mean here? Is it some sort of complex form? How should we conclude that $\frac{|dz|}{1-|z|^2}= \frac{|dw|}{1-|w|^2}$ by letting $z_1$ tends to $z_2$.
Is $|dz| = (dx)^2+(dy)^2$ which is the usual Riemannian metric. I feel that to understand the claim, one needs some rigorous treatment of the complex form maybe? Is there any reference on this matter?
More details of the claim can be found in Ahlfors, 'Conformal Invariants, Topics in Geometric Function Theory' page 2.
Questions (1) and (2) have been addressed in the comments, but I am still confused about the procedure of letting '$z_1$ approach $z_2$', since it seems to be a rather vague idea, how do you really formalize this idea to arrive the conclusion such that the given metric is invariant under the conformal mapping?