Vtyjkyuk

I'm trying to understand the proof of König's theorem in set theory using Hrbacek's and Jech's book "Introduction to Set Theory".

In the end, the proof сomes down to a certain result or lemma, which I'm having a trouble with understanding.

Let $(Y_i)_i in I$ be indexed family. Then there can be no indexed family $(Z_i)_i in I$ so that

• $(forall i,j in I)(i neq j Longrightarrow Z_icap Z_j = varnothing),$




• $(forall i in I)(|Z_i| < |Y_i|),$




• $bigcup_i in I Z_i = prod_i in I Y_i$.

Hrbacek and Jech proceed the following way. They defined a new family $(A_i)_i in I$ so that $A_i$ is the subset of $Y_i$ of all elements $y$ for which there is a function $fcolon Itobigcup_i in I Y_i$ in $Z_i$ so that $y = f(i)$. Then they claim that for any $i in I$, $A_i subsetneq Y_i$ since $|A_i| leq |Z_i| < |Y_i|$. This is where is stumble: I don't understand why we have $|A_i| leq |Z_i|$ for each $i in I$.

For each $yin A_i,$ by definition, there is a function $fin Z_i$ such that $y=f(i);$ choose such an $f$ and call it $f_y.$ (Axiom of choice used here.) The inequality $|A_i|le|Z_i|$ follows from the fact that $ymapsto f_y$ is an injective map from $A_i$ to $Z_i.$ Namely, if $y,zin A_i$ and $f_y=f_z,$ then $y=f_y(i)=f_z(i)=z.$

So, in a formula: $A_i=yin Y_i:(exists fin Z_i)(y=f(i))$, or a bit shorter: $A_i=f(i): fin Z_i$. The last formulation shows that $A_i$ is the image of $Z_i$ under the map $fmapsto f(i)$. In general: if $g:Pto Q$ is any map then $|g[P]|le|P|$ (this uses a bit of AC).