Theorem fococnv2

Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015)

	Ref	Expression
Assertion	fococnv2	$$⊢F:A\underset{onto}{⟶}B\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_B$$

Proof

Step	Hyp	Ref	Expression
1		fofun	$$⊢F:A\underset{onto}{⟶}B\to \mathrm{Fun}F$$
2		funcocnv2	$$⊢\mathrm{Fun}F\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$
3	1 2	syl	$$⊢F:A\underset{onto}{⟶}B\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$
4		forn	$$⊢F:A\underset{onto}{⟶}B\to \mathrm{ran}F=B$$
5	4	reseq2d	$$⊢F:A\underset{onto}{⟶}B\to {\mathrm{I}\upharpoonright }_{\mathrm{ran}F}={\mathrm{I}\upharpoonright }_B$$
6	3 5	eqtrd	$$⊢F:A\underset{onto}{⟶}B\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_B$$