Description: The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013)
Ref | Expression | ||
---|---|---|---|
Assertion | dmfi | ⊢ ( 𝐴 ∈ Fin → dom 𝐴 ∈ Fin ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fidomdm | ⊢ ( 𝐴 ∈ Fin → dom 𝐴 ≼ 𝐴 ) | |
2 | domfi | ⊢ ( ( 𝐴 ∈ Fin ∧ dom 𝐴 ≼ 𝐴 ) → dom 𝐴 ∈ Fin ) | |
3 | 1 2 | mpdan | ⊢ ( 𝐴 ∈ Fin → dom 𝐴 ∈ Fin ) |