Theorem f1ococnv2

Description: The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003) (Proof shortened by Stefan O'Rear, 12-Feb-2015)

	Ref	Expression
Assertion	f1ococnv2	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( 𝐹 ∘ ◡ 𝐹 ) = ( I ↾ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		f1ofo	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 : 𝐴 –onto→ 𝐵 )
2		fococnv2	⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( 𝐹 ∘ ◡ 𝐹 ) = ( I ↾ 𝐵 ) )
3	1 2	syl	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( 𝐹 ∘ ◡ 𝐹 ) = ( I ↾ 𝐵 ) )