How do you express that you successfully catch a person and prevent him from falling down?
For example, she slipped and fell down but I grabbed her
When we say "she fell", does it mean that she was already on the floor and the incident was actually happened.
But I want to express that I grabbed her just in time before hear head hit the floor.
There is no specific way of describing this situation: you just describe what you prevented from happening. If she remained standing due to your assistance, you would say:
She slipped on the wet floor, but I caught her arm and stopped/prevented her from falling.
If she did actually end up on the floor, but you prevented her head from hitting the ground, you would say:
She fainted: I caught her arm, but I wasn't fast enough to prevent her from falling. I did manage to stop/prevent her from hitting her head the ground.
Falling doesn't mean landing. So you can say "she fell but I caught her".
In your example you can say "she slipped and fell, but I grabbed her". The "and fell" is almost redundant, and could be omitted, but redundancy is very common. Its natural either way. "I caught her" sounds better than "I grabbed her".
"She fell down" or "she fell", means the action of falling is completed therefore if she fell she must have already landed on the floor. Then you can for instance, "She fell down but I quickly gave her a hand"
If you caught the person before the action of falling was not completed you can say:
She was about to fall but I caught her. I caught her before she fell. She lost her balance, but I caught her before she fell.
