Description: A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | residfi | |- ( ( _I |` A ) e. Fin <-> A e. Fin ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmresi | |- dom ( _I |` A ) = A |
|
2 | dmfi | |- ( ( _I |` A ) e. Fin -> dom ( _I |` A ) e. Fin ) |
|
3 | 1 2 | eqeltrrid | |- ( ( _I |` A ) e. Fin -> A e. Fin ) |
4 | funi | |- Fun _I |
|
5 | funfn | |- ( Fun _I <-> _I Fn dom _I ) |
|
6 | 4 5 | mpbi | |- _I Fn dom _I |
7 | resfnfinfin | |- ( ( _I Fn dom _I /\ A e. Fin ) -> ( _I |` A ) e. Fin ) |
|
8 | 6 7 | mpan | |- ( A e. Fin -> ( _I |` A ) e. Fin ) |
9 | 3 8 | impbii | |- ( ( _I |` A ) e. Fin <-> A e. Fin ) |