Theorem dmfi

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 )

Proof

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 )