Theorem fcoi2

Description: Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Assertion	fcoi2	$$⊢F:A⟶B\to \left({\mathrm{I}\upharpoonright }_B\right)\circ F=F$$

Proof

Step	Hyp	Ref	Expression
1		df-f	$$⊢F:A⟶B\leftrightarrow \left(FFnA\wedge \mathrm{ran}F\subseteq B\right)$$
2		cores	$$⊢\mathrm{ran}F\subseteq B\to \left({\mathrm{I}\upharpoonright }_B\right)\circ F=\mathrm{I}\circ F$$
3		fnrel	$$⊢FFnA\to \mathrm{Rel}F$$
4		coi2	$$⊢\mathrm{Rel}F\to \mathrm{I}\circ F=F$$
5	3 4	syl	$$⊢FFnA\to \mathrm{I}\circ F=F$$
6	2 5	sylan9eqr	$$⊢\left(FFnA\wedge \mathrm{ran}F\subseteq B\right)\to \left({\mathrm{I}\upharpoonright }_B\right)\circ F=F$$
7	1 6	sylbi	$$⊢F:A⟶B\to \left({\mathrm{I}\upharpoonright }_B\right)\circ F=F$$