Theorem f1oi

Description: A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998) (Proof shortened by Andrew Salmon, 22-Oct-2011) Avoid ax-12 . (Revised by TM, 10-Feb-2026)

	Ref	Expression
Assertion	f1oi	$$⊢\left({\mathrm{I}\upharpoonright }_A\right):A\underset{1-1\; onto}{⟶}A$$

Proof

Step	Hyp	Ref	Expression
1		fnresi	$$⊢\left({\mathrm{I}\upharpoonright }_A\right)FnA$$
2		funi	$$⊢\mathrm{Fun}\mathrm{I}$$
3		cnvi	$$⊢{\mathrm{I}}^{-1}=\mathrm{I}$$
4	3	funeqi	$$⊢\mathrm{Fun}{\mathrm{I}}^{-1}\leftrightarrow \mathrm{Fun}\mathrm{I}$$
5	2 4	mpbir	$$⊢\mathrm{Fun}{\mathrm{I}}^{-1}$$
6		funres11	$$⊢\mathrm{Fun}{\mathrm{I}}^{-1}\to \mathrm{Fun}{\left({\mathrm{I}\upharpoonright }_A\right)}^{-1}$$
7	5 6	ax-mp	$$⊢\mathrm{Fun}{\left({\mathrm{I}\upharpoonright }_A\right)}^{-1}$$
8		rnresi	$$⊢\mathrm{ran}\left({\mathrm{I}\upharpoonright }_A\right)=A$$
9		dff1o2	$$⊢\left({\mathrm{I}\upharpoonright }_A\right):A\underset{1-1\; onto}{⟶}A\leftrightarrow \left(\left({\mathrm{I}\upharpoonright }_A\right)FnA\wedge \mathrm{Fun}{\left({\mathrm{I}\upharpoonright }_A\right)}^{-1}\wedge \mathrm{ran}\left({\mathrm{I}\upharpoonright }_A\right)=A\right)$$
10	1 7 8 9	mpbir3an	$$⊢\left({\mathrm{I}\upharpoonright }_A\right):A\underset{1-1\; onto}{⟶}A$$