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I'm wondering if it's right (and not abusive or ugly) the use of two, or more, "such that" in a definition, and in the dealing with mathematical objects. I know that I could find equivalences for such definitions, and some of them more beautiful and readable; but I want to know if such use is correct.

As an example: Let A, B be sets.

$\mathcal{F}(A;B):=\Big\{f: \Big(f\subset A\times B : \; [(a,b)\in f \wedge (a,c)\in f] \Leftrightarrow b=c \Big)\Big\}$.

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    Can you give us an example of what you have in mind?2012-01-08
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    Write that as $\mathcal{F}(A;B):=\Big\{f\subset A\times B : \; [(a,b)\in f \wedge (a,c)\in f] \Leftrightarrow b=c \Big\}$. The first "$f:{}$" does not add absolutely nothing to the meaning of what you wrote!2012-01-08
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    Mariano, but if I have to use two such thats, would this usage be wrong or inappropriate?2012-01-08
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    What you've written above is syntactically incorrect (not just abusive or ugly). Set builder notation only has 1 "such that" $\{ thing_1 \;|\; thing_2 \}$.2012-01-08
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    In fact, if you're feeling really picky, set builder notation should be $\{ x \in \mathrm{Set} \;|\; \mathrm{Conditions\;placed\;on\;} x\}$2012-01-08
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    Thanks Bill. Is this because the axiom of specification?2012-01-08
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    Did you mean something like: Let $f$ be a function such that $(\forall x,y)$ $x=y$ $\Rightarrow$ $f(x)=f(y)$. This could be rewritten as: Let $f$ be such that $f(x)=f(y)$ for any $x$, $y$ such that $x=y$. (Here I used "such that" twice.) Or: Let $f$ be such that $f(x)=f(y)$ whenever $x=y$. (Here I avoided using "such that" twice.) Of course this example is rather artificial - we would simply say: Let $f$ be injective.2012-01-08
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    Yes Martin, something like that to.2012-01-08
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    Your first colon is a delimiter character in the [set builder notation](http://en.wikipedia.org/wiki/Set-builder_notation) - often pronounced "such that". Your second colon is not part of any set-builder notation. It appears that you intend it to denote the words "such that". Here, instead of this colon, you should use $\:\wedge\:$ (logical "and"), as you do elsewhere. Or you could move the universe specification to the left of the colon, as Mariano remarked.2012-01-08

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In my paper Canonical seeds and Prikry trees, I had a need to consider the iterated ultrapower arising from a product measure $\mu=\nu\times\eta$, and so it was necessary to consider $\mu\times\mu$, a product of products. If you look on page 394, you will see that I considered $A\in f\ast\mu^2$, where $f(\langle w,x\rangle,\langle y,z\rangle)=\langle w,z\rangle$ is the mixed projection function, and I proceeded to argue that $$\Bigl\{w\mid \Bigl\{x\mid \{y\mid \{z\mid \langle w,z\rangle\in A\strut\}\in\eta\}\in\nu\Bigr\}\in\eta\Bigr\}\in\nu.$$

Although I won't speak to whether this notation is abusive or ugly, nevertheless I do find that it succinctly and exactly expresses the necessary fact at the heart of the matter, and so I used it.

(This fact also appears in my dissertation, from part of which this paper was adapted.)