If the number of functions from $A \to B$ is equal to $A$, then the cardinality of A and B is $1$.   In set theory notation: If $|B|^{|A|}=|A|$ then $|A|=|B|=1$.  Any ideas? I was think of assuming $|A|$ or $|B| \neq 1$ but can't think of anything I can do after/
Show (within ZF Set Theory without choice) that for any sets $A,B,C:$   If $|A|^{|B|} = |B|$, then $|A| = 1 = |B|.$   That is, if there is a bijection from $|A|^{|B|}$ to $|B|$, then  $|A| = 1 = |B|.$  Please help me..
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