What is the name of the vertical bar in $(x^2+1)\vert_{x = 4}$ or $\left.\left(\frac{x^3}{3}+x+c\right) \right\vert_0^4$?

I've always wanted to know what the name of the vertical bar in these examples was:

$f(x)=(x^2+1)\vert_{x = 4}$ (I know this means evaluate $x$ at $4$)

$\int_0^4 (x^2+1) \,dx = \left.\left(\frac{x^3}{3}+x+c\right) \right\vert_0^4$ (and I know this means that you would then evaluate at $x=0$ and $x=4$, then subtract $F(4)-F(0)$ if finding the net signed area)

I know it seems trivial, but it's something I can't really seem to find when I go googling and the question came up in my calc class last night and no one seemed to know.

Also, for bonus internets; What is the name of the horizontal bar in $\frac{x^3}{3}$? Is that called an obelus?

Jeff Miller calls it "bar notation" in his Earliest Uses of Symbols of Calculus (see below). The bar denotes an evaluation functional, a concept whose importance comes to the fore when one studies duality of vector spaces (e.g. such duality plays a key role in the Umbral Calculus).

The bar notation to indicate evaluation of an antiderivative at the two limits of integration was first used by Pierre Frederic Sarrus (1798-1861) in 1823 in Gergonne’s Annales, Vol. XIV. The notation was used later by Moigno and Cauchy (Cajori vol. 2, page 250).

Below is the cited passage from Cajori

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This may be called Evaluation bar. See, in particular, here (Evaluation Bar Notation:).

In my calculus book, the vertical bar is called the "evaluation symbol", and this phrase is bolded when first mentioned. It makes sense, I suppose.

Copy paste from wikipedia: Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a vinculum or fraction bar, between them.

In the wikipedia article for the symbol no name for this particular use of it is mentioned, just that it is read as, simply, "evaluated at". It has a number of suggested names for the symbol from different situations though: