Theorem f1f1orn

Description: A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004)

	Ref	Expression
Assertion	f1f1orn	$$⊢F:A\underset{1-1}{⟶}B\to F:A\underset{1-1\; onto}{⟶}\mathrm{ran}F$$

Proof

Step	Hyp	Ref	Expression
1		f1fn	$$⊢F:A\underset{1-1}{⟶}B\to FFnA$$
2		df-f1	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \mathrm{Fun}{F}^{-1}\right)$$
3	2	simprbi	$$⊢F:A\underset{1-1}{⟶}B\to \mathrm{Fun}{F}^{-1}$$
4		f1orn	$$⊢F:A\underset{1-1\; onto}{⟶}\mathrm{ran}F\leftrightarrow \left(FFnA\wedge \mathrm{Fun}{F}^{-1}\right)$$
5	1 3 4	sylanbrc	$$⊢F:A\underset{1-1}{⟶}B\to F:A\underset{1-1\; onto}{⟶}\mathrm{ran}F$$