It is difficult to give a definite answer to your questions because definitions and theorem orders differ. But generally you work is correct. You have shown that a line segment is mapped into an line segment. You may want to expand to include the entire line.

Recall that if three points, P Q & X, are collinear then P-X-Q, P-Q-X, or X-P-Q.

You have done the first, the segment. The others follow from what you have done.

But again, you must follow your instructor’s standards for rigor.