I will come back later and re-read your proof but you're idea is correct.Right so, I have this major block with set exponentiation and unfortunately we have to do it in my set theory course with cardinals.
I'm gonna post the question and the answer and post what (after many months of thinking about it) is the best I can do on my own...
Question. If κ, λ and μ are cardinals with 0 < κ ≤ λ. Show that
Solution. Let κ = P(A), λ = P(B) and μ = P(C). Let f:A→ B be an injection.
Let k:B→ A be the surjective map defined in the proof of Prop 1.3.1 which exists because κ > 0.
Then we define a map by G(g)(b) = g(k(b)) (+++).
We claim G is injective.
If G(g)=G(g′) then g(k(b))=g′(k(b)).
Since k is surjective k(b) ranges over all possible elements of A and so g(a)=g′(a) for all a ∈ A. Hence, g=g′
Notice the (+++), this is the part that scrambles my brain, making injections, surjections and bijections.
I'm not quite sure what is either, I seem to think it's a possible injection that takes an element in to .
But, my attempted answer is slightly different and I want to know if it looks mildly right and if not, can you explain what the map in (+++) actually means!
So we have is an injective map.
Define as a surjective map from B to A.
So, for every there is a such that .
So, now the bit (for me)... We define a map from by...
I think I'll just stop there though, the rest of the Q is a bit easier, just the defining the map part that gets me.
Having just re-read it I'm pretty sure mine isn't even a map, it's just a renaming of .