Description: If the function's value at an argument is not the empty set, it equals the alternate function value at this argument. (Contributed by AV, 3-Sep-2022)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | afv2fvn0fveq | |- ( ( F ` A ) =/= (/) -> ( F '''' A ) = ( F ` A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvfundmfvn0 | |- ( ( F ` A ) =/= (/) -> ( A e. dom F /\ Fun ( F |` { A } ) ) ) |
|
| 2 | df-dfat | |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) ) |
|
| 3 | 1 2 | sylibr | |- ( ( F ` A ) =/= (/) -> F defAt A ) |
| 4 | dfatafv2eqfv | |- ( F defAt A -> ( F '''' A ) = ( F ` A ) ) |
|
| 5 | 3 4 | syl | |- ( ( F ` A ) =/= (/) -> ( F '''' A ) = ( F ` A ) ) |