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Metamath Proof Explorer
Description: If an onto function has a finite domain, its codomain/range is finite.
Theorem 37 of Suppes p. 104. (Contributed by NM, 25-Mar-2007)
|
|
Ref |
Expression |
|
Assertion |
fofi |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) → 𝐵 ∈ Fin ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fodomfi |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) → 𝐵 ≼ 𝐴 ) |
| 2 |
|
domfi |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴 ) → 𝐵 ∈ Fin ) |
| 3 |
1 2
|
syldan |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) → 𝐵 ∈ Fin ) |