Statement: For each $x$ in $X$, there exists a unique $y$ in $Y$ such that $(x,y)\in f$.
Which is the correct translation?
- $\forall x\exists !y(x\in X\rightarrow (y\in Y\land (x,y)\in f)$
- $\forall x(x\in X\rightarrow \exists !y(y\in Y\land (x,y)\in f)$
- $\forall x\exists y(x\in X\rightarrow (y\in Y\land (x,y)\in f)\land\forall x\forall y\forall z((x,y)\in f\land (x,z)\in f\rightarrow y=z)$
Are any of these equivalent? Also please consider the case when $Y$ is not singleton.
You might want to have a look at this question.