Theorem f1cocnv1

Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011)

	Ref	Expression
Assertion	f1cocnv1	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		f1f1orn	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 )
2		f1ococnv1	⊢ ( 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐴 ) )
3	1 2	syl	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ◡ 𝐹 ∘ 𝐹 ) = ( I ↾ 𝐴 ) )