Theorem f1cocnv1

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

	Ref	Expression
Assertion	f1cocnv1	$$⊢F:A\underset{1-1}{⟶}B\to F^{-1}\circ F={\mathrm{I}\upharpoonright }_A$$

Proof

Step	Hyp	Ref	Expression
1		f1f1orn	$$⊢F:A\underset{1-1}{⟶}B\to F:A\underset{1-1\; onto}{⟶}\mathrm{ran}F$$
2		f1ococnv1	$$⊢F:A\underset{1-1\; onto}{⟶}\mathrm{ran}F\to F^{-1}\circ F={\mathrm{I}\upharpoonright }_A$$
3	1 2	syl	$$⊢F:A\underset{1-1}{⟶}B\to F^{-1}\circ F={\mathrm{I}\upharpoonright }_A$$