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	$$⊢F:A\underset{1-1\; onto}{⟶}B\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_B$$

Proof

Step	Hyp	Ref	Expression
1		f1ofo	$$⊢F:A\underset{1-1\; onto}{⟶}B\to F:A\underset{onto}{⟶}B$$
2		fococnv2	$$⊢F:A\underset{onto}{⟶}B\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_B$$
3	1 2	syl	$$⊢F:A\underset{1-1\; onto}{⟶}B\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_B$$