Theorem funcocnv2

Description: Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Assertion	funcocnv2	$$⊢\mathrm{Fun}F\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$

Proof

Step	Hyp	Ref	Expression
1		df-fun	$$⊢\mathrm{Fun}F\leftrightarrow \left(\mathrm{Rel}F\wedge F\circ F^{-1}\subseteq \mathrm{I}\right)$$
2	1	simprbi	$$⊢\mathrm{Fun}F\to F\circ F^{-1}\subseteq \mathrm{I}$$
3		iss	$$⊢F\circ F^{-1}\subseteq \mathrm{I}\leftrightarrow F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{dom}\left(F\circ F^{-1}\right)}$$
4		dfdm4	$$⊢\mathrm{dom}F=\mathrm{ran}{F}^{-1}$$
5		dmcoeq	$$⊢\mathrm{dom}F=\mathrm{ran}{F}^{-1}\to \mathrm{dom}\left(F\circ F^{-1}\right)=\mathrm{dom}{F}^{-1}$$
6	4 5	ax-mp	$$⊢\mathrm{dom}\left(F\circ F^{-1}\right)=\mathrm{dom}{F}^{-1}$$
7		df-rn	$$⊢\mathrm{ran}F=\mathrm{dom}{F}^{-1}$$
8	6 7	eqtr4i	$$⊢\mathrm{dom}\left(F\circ F^{-1}\right)=\mathrm{ran}F$$
9	8	reseq2i	$$⊢{\mathrm{I}\upharpoonright }_{\mathrm{dom}\left(F\circ F^{-1}\right)}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$
10	9	eqeq2i	$$⊢F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{dom}\left(F\circ F^{-1}\right)}\leftrightarrow F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$
11	3 10	bitri	$$⊢F\circ F^{-1}\subseteq \mathrm{I}\leftrightarrow F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$
12	2 11	sylib	$$⊢\mathrm{Fun}F\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$