Suppose we have a function $$ f(x)= x+1 $$ for $0<x<1$. Introduce $y=x^3$ for $0<y<1$. So in $y$ space we would write $$ f(y)= \sqrt[3]{y}+1. $$ But why is that? Why isn't it just $$ f(y) = y+1? $$ What I am asking is, in the first equation we have a formula for what happens when you pass an argument to the function, why can't we apply that to $y$? I would think that the underlying reason is that in the first case we are mapping from $X$ space and from the second we are mapping from $Y$ space, but here even the domains are the same.
For some reason this is very confusing to me. I have always found this award even when doing differential equations by substitution.
Could someone clarify this for me?
the formula reads:
$(\forall x)(\forall y)[x,y\in(0,1)\land (f(x)=(x+1))\land y=x^3)]$
So $x,y$ are bound variables.
I.e. $x,y$ are in the scope of the quantifiers $(\forall x), (\forall y)$, they appear in the first formula following the quantifiers.
This is why you make the first substitution.