Theorem finresfin

Description: The restriction of a finite set is finite. (Contributed by Alexander van der Vekens, 3-Jan-2018)

	Ref	Expression
Assertion	finresfin	⊢ ( 𝐸 ∈ Fin → ( 𝐸 ↾ 𝐵 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		resss	⊢ ( 𝐸 ↾ 𝐵 ) ⊆ 𝐸
2		ssfi	⊢ ( ( 𝐸 ∈ Fin ∧ ( 𝐸 ↾ 𝐵 ) ⊆ 𝐸 ) → ( 𝐸 ↾ 𝐵 ) ∈ Fin )
3	1 2	mpan2	⊢ ( 𝐸 ∈ Fin → ( 𝐸 ↾ 𝐵 ) ∈ Fin )