Putting "$\forall y(y \in x \to \exists A \in F(y \in A))$" into words

I'm new to mathematical proof and I struggle sometimes with putting definitions into words. If I had one like this:

$$\forall y(y \in x \to \exists A \in F(y \in A))$$

Would it be correct to read this as follows?

For all y such that if y is an element of x, then there exists a set A in a family of sets F such that y is an element of A.

Solution 1:

Compare:

• For each ball such that if it is green, then it is heavy.    (incoherent)
• For each ball such that if it is green, it is heavy.    (incoherent)
• For each ball, if it is green, then it is heavy.    (OK)
• For each ball, if it is green, it is heavy.    (OK)
• For each ball such that it is green, it is heavy.    (OK)
• For each ball that is green, it is heavy.    (better)
• For each green ball, it is heavy.    (even better)
• Every green ball is heavy.    (best)

Viewing the first bullet point as a chopped-off sentence, adding parentheses to clarify its structure, and completing it:

• For each ball such that (if it is green, then it is heavy), it is smooth.   (now coherent)

$$\forall y\,\big(y \in x \to \exists A {\in} F\,(y \in A)\big)$$

Would it be correct to read this as follows?

For all $y$ such that if $y$ is an element of $x,$ then there exists a set $A$ in a family of sets $F$ such that y is an element of $A.$

As illustrated above, your suggested reading becomes grammatical once “if...then” or “such that” is dropped. Let's correct it by replacing “such that” with a comma:

• For all $y,$ if $y$ is an element of $x,$ then there exists a set $A$ in a family of sets $F$ such that $y$ is an element of $A.$

This literal reading can be condensed in several ways, the most succinctly as suggested by Pilcrow:

• Each element of $\color{red}x$ lies in some set in $\color{green}F.$

Abbreviating and rewriting the given formula: $$∀y{\in}\color{red}x\;∃A{\in}\color{green}F\;y\in A\\∀p{\in}\color{red}x\;∃q{\in}\color{green}F\;p\in q.$$