Theorem f1ores

Description: The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998)

	Ref	Expression
Assertion	f1ores	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ ( 𝐹 “ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		f1ssres	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐵 )
2		f1f1orn	⊢ ( ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐵 → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ ran ( 𝐹 ↾ 𝐶 ) )
3	1 2	syl	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ ran ( 𝐹 ↾ 𝐶 ) )
4		df-ima	⊢ ( 𝐹 “ 𝐶 ) = ran ( 𝐹 ↾ 𝐶 )
5		f1oeq3	⊢ ( ( 𝐹 “ 𝐶 ) = ran ( 𝐹 ↾ 𝐶 ) → ( ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ ( 𝐹 “ 𝐶 ) ↔ ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ ran ( 𝐹 ↾ 𝐶 ) ) )
6	4 5	ax-mp	⊢ ( ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ ( 𝐹 “ 𝐶 ) ↔ ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ ran ( 𝐹 ↾ 𝐶 ) )
7	3 6	sylibr	⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1-onto→ ( 𝐹 “ 𝐶 ) )