reference request: convert minimax into a unique restricted minimization

I have a minimax problem that I want to turn into a single minimization problem. I wonder if the approach is standard and, if so, if someone can point me to a reference.

Consider the problem

$$ V (x) = min_ {y} max {f (x, y), g (x, y) } $$

To do this, consider the reformulation of the internal maximization problem

$$ min_z {z } text {subject to} f (x, y) le z, g (x, y) le z $$

Taking a partial Lagrangian, the above can be written as

$$ min_z {z + lambda (g (x, y) -z) } text {subject to} f (x, y) le z $$

So the problem is equivalent.

$$ V (x) = min _ { lambda ge0, {z: z ge f (x, y) }, y} {z + lambda (g (x, y) -z) } $$

Question: Is the above correct? If so, is it standard (reference)?