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 | |- ( _I |` A ) : A -1-1-onto-> A |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnresi | |- ( _I |` A ) Fn A |
|
| 2 | funi | |- Fun _I |
|
| 3 | cnvi | |- `' _I = _I |
|
| 4 | 3 | funeqi | |- ( Fun `' _I <-> Fun _I ) |
| 5 | 2 4 | mpbir | |- Fun `' _I |
| 6 | funres11 | |- ( Fun `' _I -> Fun `' ( _I |` A ) ) |
|
| 7 | 5 6 | ax-mp | |- Fun `' ( _I |` A ) |
| 8 | rnresi | |- ran ( _I |` A ) = A |
|
| 9 | dff1o2 | |- ( ( _I |` A ) : A -1-1-onto-> A <-> ( ( _I |` A ) Fn A /\ Fun `' ( _I |` A ) /\ ran ( _I |` A ) = A ) ) |
|
| 10 | 1 7 8 9 | mpbir3an | |- ( _I |` A ) : A -1-1-onto-> A |