What is the complete proof that the speed of light in vacuum is constant in relativistic mechanics?

In the main principles I've read that the speed of light is constant since we can calculate it from the Maxwell equations.

The fact that the speed of light could be deduced from Maxwell's equations does not, in and of itself, imply that the speed of light is constant in all reference frames. Certainly the equations don't make an obvious reference to a reference frame; but once you've made the connection between electric and magnetic fields and light, it seems pretty obvious what the "natural" rest frame is (bolding mine):

We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.

– James Clerk Maxwell, On the Physical Lines of Force

In other words, one could easily imagine a world in which Maxwell's equations are only valid in the rest frame of the luminiferous aether — and from about 1860–1905 or so, this is precisely the universe that physicists thought we lived in. In such a universe, Maxwell's equations would in fact look different in different reference frames; a "full" version of these equations would include terms that depended on an observer's velocity $\vec{v}$ with respect to the aether. There is nothing mathematically inconsistent about the equations describing such a Universe.

What these equations are inconsistent with, however, is two things: (1) experimental evidence, and (2) our sense of symmetry. The Michelson-Morley experiment was designed to detect Earth's motion relative to the aether — in other words, to indirectly verify the presence of these $\vec{v}$-dependent terms in the hypothetical Maxwell's equations. Of course, they famously came up short.

The other problem is that there seem to be a lot of convenient coincidences between what seem to be the same phenomena described in different reference frames:

It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest.

— Albert Einstein, On the Electrodynamics of Moving Bodies

Or, to summarize: if I move a coil near a magnet, the magnetic field causes the charges to flow. If I move a magnet near a coil, the changing magnetic field causes an electric field, which causes the charges to flow. These two descriptions seem very different, and yet somehow they give rise to exactly the same amount of current in the coil. Einstein's contention was that this couldn't be a coincidence, and that only relative velocity should matter.

If you buy that, then you find (as Einstein did) that when you go into another reference frame, the electric and magnetic fields intermingle with each other. If you look at the above link to Einstein's original paper, §6 describes how the electric and magnetic fields transform into each other. His notation is a little antiquated — what he calls $(X, Y, Z)$ we would nowadays usually call $(E_x, E_y, E_z)$, and what he calls $(L, M, N)$ we would usually call $(B_x, B_y, B_z)$. In different reference frames moving relative to each other in the $x$-direction, all of these components change, and the components $E_y$, $E_z$, $B_y$, and $B_z$ get mixed up with each other. In other words, the electric and magnetic field strengths observed by Observer A and Observer B are not necessarily the same.

These transformations between the fields are a necessary consequence of the postulate that the laws of physics are the same in all reference frames. But Maxwell's equations don't necessarily imply that the laws of physics are all the same in such reference frames; they are agnostic on the subject. Historically, physicists originally believed that there was in fact a privileged frame in which Maxwell's equations held exactly, and it was only after careful experimentation and careful thought that we figured out that Maxwell's equations were also consistent with the principle of relativity.

It's an axiom based on observation. As far as I know, that means it cannot be proved.

Also, Maxwell's equations are relativistically covariant. The thing you have to keep in mind is that the electric and magnetic field get mixed by Lorentz transformations. Think of the force law on a moving charge:

$$\mathbf{F} = q \mathbf{E} + q\mathbf{v}\times \mathbf{B}.$$

Now, imagine there's a charge moving at speed $\mathbf{v}$ through some magnetic field, and there's no electric field. You'll see the charge accelerate under the influence of the magnetic force, curving it's path. Now, imagine what an observer also moving at $\mathbf{v}$, instantaneously, sees. That observer sees an instantaneously stationary charge, so it can't experience a magnetic force. That observer has to see the charge accelerate in some fashion, though, because you saw it accelerate. That can only be the case if that observer thinks that there are both electric and magnetic fields present.

As mentioned in other answers, Maxwell's equations are indeed invariant under Lorentz transformations. Easiest way to see this is to write them in a manifestly covariant form. This form is: \begin{align} \partial_\mu F^{\mu\nu} &= j^\nu \\ \partial^\tau F^{\mu\nu}+\partial^\mu F^{\nu\tau} + \partial^\nu F^{\tau\mu}&=0 \end{align} where $F^{\mu\nu}$ is electromagnetic tensor (which contains electric and magnetic field in a 4x4 matrix), $j^\mu$ is the four-current (which contains the charge density and the current density in a four-vector) and natural units $\epsilon_0 = \mu_0 = c=1$ are used for the sake of simplicity.

Electromagnetic tensor $F^{\mu\nu}$ is given in terms of four-potential $A^\mu$ (which contains electric (scalar) and magnetic (vector) potential) is related to $F^{\mu\nu}$ via \begin{equation} F^{\mu\nu}=\partial^\mu A^\nu - \partial^\nu A^\mu \end{equation} and when you plug it in $\partial_\mu F^{\mu\nu} = j^\nu$ without sources ($j^\mu=0$), you get \begin{equation} \Box A^\mu-\partial^\mu(\partial\cdot A)=0. \end{equation} However, there are multiple choices of $A^\mu$ which give the same $F^{\mu\nu}$. Transformation $A^\mu \to A^\mu+\partial^\mu \xi$, where $\xi$ is a scalar function, leaves $F^{\mu\nu}$ unchanged and it is called "gauge transformation". If you take the original $A^\mu$ and make a gauge transformation with $\xi=\partial\cdot A$, you get something called "Lorentz gauge", characterized by $\partial\cdot A=0$. In the Lorentz gauge, the above equation simplifies to \begin{equation} \Box A^\mu=0 \end{equation} which is a wave equation for waves moving at the speed $c$. Therefore, from Maxwell's equations follows that electromagnetic radiation, including light, travels at the speed $c$ in vacuum.