Description: Characterization of the elements of a restricted identity relation. (Contributed by BJ, 28-Aug-2022) (Proof shortened by Peter Mazsa, 9-Sep-2022)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | elrid | |- ( A e. ( _I |` X ) <-> E. x e. X A = <. x , x >. ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-res | |- ( _I |` X ) = ( _I i^i ( X X. _V ) ) |
|
| 2 | 1 | eleq2i | |- ( A e. ( _I |` X ) <-> A e. ( _I i^i ( X X. _V ) ) ) |
| 3 | elidinxp | |- ( A e. ( _I i^i ( X X. _V ) ) <-> E. x e. ( X i^i _V ) A = <. x , x >. ) |
|
| 4 | inv1 | |- ( X i^i _V ) = X |
|
| 5 | 4 | rexeqi | |- ( E. x e. ( X i^i _V ) A = <. x , x >. <-> E. x e. X A = <. x , x >. ) |
| 6 | 2 3 5 | 3bitri | |- ( A e. ( _I |` X ) <-> E. x e. X A = <. x , x >. ) |