Description: Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019) Remove extra hypothesis. (Revised by SN, 17-Apr-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rnmptc.f | |- F = ( x e. A |-> B ) |
|
| rnmptc.a | |- ( ph -> A =/= (/) ) |
||
| Assertion | rnmptc | |- ( ph -> ran F = { B } ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rnmptc.f | |- F = ( x e. A |-> B ) |
|
| 2 | rnmptc.a | |- ( ph -> A =/= (/) ) |
|
| 3 | fconstmpt | |- ( A X. { B } ) = ( x e. A |-> B ) |
|
| 4 | 1 3 | eqtr4i | |- F = ( A X. { B } ) |
| 5 | 4 | rneqi | |- ran F = ran ( A X. { B } ) |
| 6 | rnxp | |- ( A =/= (/) -> ran ( A X. { B } ) = { B } ) |
|
| 7 | 5 6 | syl5eq | |- ( A =/= (/) -> ran F = { B } ) |
| 8 | 2 7 | syl | |- ( ph -> ran F = { B } ) |