Theorem funcocnv2

Description: Precomposition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by BJ, 25-Oct-2026)

	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	2 3	sylib	$$⊢\mathrm{Fun}F\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{dom}\left(F\circ F^{-1}\right)}$$
5		dmcocnv2	$$⊢\mathrm{dom}\left(F\circ F^{-1}\right)=\mathrm{ran}F$$
6	5	reseq2i	$$⊢{\mathrm{I}\upharpoonright }_{\mathrm{dom}\left(F\circ F^{-1}\right)}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$
7	4 6	eqtrdi	$$⊢\mathrm{Fun}F\to F\circ F^{-1}={\mathrm{I}\upharpoonright }_{\mathrm{ran}F}$$