How can I prove the following statement?
Let $ X $ and $ Y $ sets and $ f_i , i \in I, $ be a family of partial functions in $ X. $ Suppose the family $ \{f_i \}_{i \in I} $ satisfies the following compatibility condition:
$ \forall \ i, j \in I, $ if $ x \in \ \mbox{dom} \ f_i \cap \mbox {dom} \ f_j, $ then $ f_i (x) = f_j (x). $
Under these conditions, there is a unique function
$$ f \colon \bigcup\limits_{i \in I} \mbox{dom} f_i \to Y $$
such that for all $ \forall i \in I \wedge \forall x \in \mbox{dom} f_i, $ we have $ f(x) = f_i (x). $
This is supposed to be obvious but I suppose it depends on your allowable definition of a function. For x in your proposed domain you must produce a unique value f(x) .Choose any i with $x\in dom f_i$ and define f(x)=$f_i(x)$ . By your hypothesis the value is independent of i .