Is ''isomorphic'' between projective planes actually equivalence relation?

Question

I'm new to projective geometry and I just read a few pages of Hartshone's Foundation of Projective Geometry. On page 5 he defines two projective planes to be isomorphic if there exists, a bijection from one to another that takes collinear points into collinear points. However, is ''isomorphic'' really an equivalence relation? i.e. does a bijection like that take collinear points back to collinear points? I didn't find out a way to prove that..

Answer 1

I think I solved it.

Suppose $f$ is a bijective mapping that takes collinear points in $S$ to collinear points in $S'$, we wish to prove that $f^{-1}$ also has this property, i.e. if $f$ takes $A, B, C$ to $A',B',C'$ which are collinear, then $A,B,C$ must be collinear:

If not, then the lines $AB$ and $BC$ intersect in only the point $B$. For any point $D$ on $AB$, since $A,B,D$ are collinear, the image of $D$ must lie on the line $A'B'C'$. For the same reason, so is the image of each point lying on the line $BC$. Since we know that the image of every point on $AB$ and $BC$ is on $A'B'C'$, we can use the property of $f$ again to see that the image of every line that intersects $AB$ and $BC$ at different points must also be on $A'B'C'$. However, through any point in $S$ except $A,B,C$, such a line exists. Hence we know that $f$ takes all the points in $S$ into the line $A'B'C'$, which is a contradiction to the fact that $f$ is a bijection, since every projective plane (here we indicate $S'$) must have points which are not collinear.