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Metamath Proof Explorer
Description: A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004)
|
|
Ref |
Expression |
|
Assertion |
funforn |
⊢ ( Fun 𝐴 ↔ 𝐴 : dom 𝐴 –onto→ ran 𝐴 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
funfn |
⊢ ( Fun 𝐴 ↔ 𝐴 Fn dom 𝐴 ) |
| 2 |
|
dffn4 |
⊢ ( 𝐴 Fn dom 𝐴 ↔ 𝐴 : dom 𝐴 –onto→ ran 𝐴 ) |
| 3 |
1 2
|
bitri |
⊢ ( Fun 𝐴 ↔ 𝐴 : dom 𝐴 –onto→ ran 𝐴 ) |