I apologize for asking yet another trivial question, but here it goes i am interested in formulating a proof to show that the set $L$ of all functions on $A$ into $B$ exists. Now as per my readings i believe proofs of existence only require us to show one such case.
Now clearly such a set L is a subset(may be a proper subset) of $P(A \times B)$.
I can create a trivial example say if $A = \{a_1, a_2\}$ and B={b1, b2} then (A X B) = {(a1, b1), (a2, b1), (a1, b2), (a2, b2)}, and the Power Set $P(A \times B)$ = {{}, {(a1, b1)}, {(a2, b1)}, {(a1, b2)}, {(a2, b2)}, ..., $A \times B$}.
Since a function is simply a binary relation such that if (a1,b1) and (a1,b2) exist in it then b1 = b2. Hence as long as we satisfy this property we can pick ordered pairs from (A X B) to form our function and various permutations of such pickings would give us our set L of our functions, so such a set L must exist ?
Would this be considered a valid existence proof ? Any help or guidance would be appreciated.