Theorem f1cocnv2

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

	Ref	Expression
Assertion	f1cocnv2	$$⊢F:A\underset{1-1}{⟶}B\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$

Proof

Step	Hyp	Ref	Expression
1		f1fun	$$⊢F:A\underset{1-1}{⟶}B\to \mathrm{Fun}F$$
2		funcocnv2	$$⊢\mathrm{Fun}F\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$
3	1 2	syl	$$⊢F:A\underset{1-1}{⟶}B\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$