This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem rnmptfi

Description: The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019)

		Ref	Expression
	Hypothesis	rnmptfi.a	$⊢ A = (x \in B ⟼ C)$
	Assertion	rnmptfi	$⊢ B \in Fin \to ran A \in Fin$

Step	Hyp	Ref	Expression
1		rnmptfi.a	$⊢ A = (x \in B ⟼ C)$
2		mptfi	$⊢ B \in Fin \to (x \in B ⟼ C) \in Fin$
3	1 2	eqeltrid	$⊢ B \in Fin \to A \in Fin$
4		rnfi	$⊢ A \in Fin \to ran A \in Fin$
5	3 4	syl	$⊢ B \in Fin \to ran A \in Fin$