Theorem f1cnv

Description: The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011)

	Ref	Expression
Assertion	f1cnv	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ◡ 𝐹 : ran 𝐹 –1-1-onto→ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		f1f1orn	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 )
2		f1ocnv	⊢ ( 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 → ◡ 𝐹 : ran 𝐹 –1-1-onto→ 𝐴 )
3	1 2	syl	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ◡ 𝐹 : ran 𝐹 –1-1-onto→ 𝐴 )