Metamath Proof Explorer

Theorem fonum

Description: A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 30-Apr-2015)

		Ref	Expression
	Assertion	fonum	$⊢ (A \in dom card \land F : A \underset{onto}{⟶} B) \to B \in dom card$

Step	Hyp	Ref	Expression
1		fodomnum	$⊢ A \in dom card \to (F : A \underset{onto}{⟶} B \to B ≼ A)$
2	1	imp	$⊢ (A \in dom card \land F : A \underset{onto}{⟶} B) \to B ≼ A$
3		numdom	$⊢ (A \in dom card \land B ≼ A) \to B \in dom card$
4	2 3	syldan	$⊢ (A \in dom card \land F : A \underset{onto}{⟶} B) \to B \in dom card$