newtonian mechanics - For what reason is the difference of potential energy $Delta U=-W$ equal to the *opposite* of the work done?

In my classical mechanics physics textbook (a translation of the Walker-Halliday-Resnick Fundamentals of Physics) the difference of potential energy is defined as

$$\Delta U = -W \qquad (1)$$

I have done extensive research (taking me 5+ hours) and I claim to have a reasonable understanding of this model. In particular, I understand that if we throw a solid object in a straight upward direction then the work (i.e., the quantity of kinetic energy conveyed or subtracted from a body) exerted by the Earth's gravitational force is negative because they act on opposite directions: $W = \vec{F} \cdot \vec{d} = F \cdot cos(\phi) \cdot d$, where $cos(\phi) = -1$ due to $\phi$, the angle between the movement and the gravitational force, being $180°$.

However, I couldn't find anywhere an explanation for this. I was demonstrated that for a conservative force $\vec{F}$ doing work along a path $ab$, $W_{ab} = -W_{ba}$, and I also know that we can always associate a potential energy to a conservative force. But I'm still missing a link, and not knowing how the negative work of a force relates to its potential energy gives me brain fog.

Can you please provide an explanation, or an appropriate proof, for $(1)$? Please note that my physics knowledge only extends up to what is taught in university-level Physics I and Physics II courses.