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Metamath Proof Explorer

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 ) )