How to derive joint CDF using definition of random variables

I am reading the article "An Introduction to Copulas" http://www.columbia.edu/~mh2078/QRM/Copulas.pdf by Martin Haugh. I’d love to have some help to understand an example (page 2) that is quoted below:
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Can anyone help me understand how the equation (3) above is derived? This is obviously a beginner question, but I feel it can be good opportunity for me to better understand some of the basic concepts. Thanks!