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Metamath Proof Explorer

Theorem abrexfi

Description: An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014)

		Ref	Expression
	Assertion	abrexfi	$⊢ A \in Fin \to \{y \| \exists x \in A y = B\} \in Fin$

Step	Hyp	Ref	Expression
1		eqid	$⊢ (x \in A ⟼ B) = (x \in A ⟼ B)$
2	1	rnmpt	$⊢ ran (x \in A ⟼ B) = \{y \| \exists x \in A y = B\}$
3		mptfi	$⊢ A \in Fin \to (x \in A ⟼ B) \in Fin$
4		rnfi	$⊢ (x \in A ⟼ B) \in Fin \to ran (x \in A ⟼ B) \in Fin$
5	3 4	syl	$⊢ A \in Fin \to ran (x \in A ⟼ B) \in Fin$
6	2 5	eqeltrrid	$⊢ A \in Fin \to \{y \| \exists x \in A y = B\} \in Fin$