The internal energy of a system can be increased in two ways. (i) By supplying heat to the system, (ii) By doing work on the system. Then just combine these values. But how do you exactly know that there will be no other terms related to this proof ?
Consider a simple block of the type we'd envision for kinematics problems in introductory physics. It has no internal structure that matters. We could give it energy by raising it or by accelerating it, and we could measure its energy from any arbitrary reference height or speed. But whatever we do, the energy is expressed as just one number: 5 J, or 0 J, or -130 J, say. That's it.
Real objects are more complex. They contain, e.g., 1025 particles (molecules), each of which has its own energy. We could never catalog each value in this energy distribution. Fortunately, the distribution is often of a consistent general shape, and we need only two numbers to characterize it: the average or center value, say, and the breadth, or amount of variation or dispersion, say. This is similar to the way we describe statistical distributions such as the Gaussian, gamma, or lognormal distributions, for instance: by a location (perhaps the mean or median) and a width (perhaps the standard deviation or 95% confidence interval). For real objects near thermodynamic equilibrium, the location concept broadly corresponds to the energy, and the dispersion concept broadly corresponds to the temperature.
Stated again: if you have not just one energy but many, for a well-understood simple distribution, then you can describe the system in terms of just the distribution location and breadth. In turn, there are two ways you can alter this distribution: by changing its position (i.e., the energies of all particles in concert) or by changing the breadth (i.e., the width or dispersion of particle energies). The former is to do work on the system, and the latter is to heat it. When we do work on a system, we elevate all particle energies together; when we heat a system, we broaden the distribution of particle energies. This is the distinction between work and heat. Does this answer your question?
(Consider, for example, two very cold flywheels rotating in opposite directions. We suddenly press them together, causing them to brake to a halt and heat up in the process. In our frame, the linear and angular momenta remain zero before and after. The total mass stays the same, as does the total energy. But we've converted two very narrow peaks of energy—the energy of every molecule could initially be classified precisely by simply "clockwise" or "counterclockwise"—to a relatively broad distribution that is now amenable to description only by the equilibrium temperature, as the distinct particle information is now too vast to track. Colloquially, the work we did to start the wheels spinning has been converted entirely to heat.)