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Metamath Proof Explorer

Theorem funmpt

Description: A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013)

		Ref	Expression
	Assertion	funmpt	$⊢ Fun (x \in A ⟼ B)$

Step	Hyp	Ref	Expression
1		funopab4	$⊢ Fun \{⟨x, y⟩ \| (x \in A \land y = B)\}$
2		df-mpt	$⊢ (x \in A ⟼ B) = \{⟨x, y⟩ \| (x \in A \land y = B)\}$
3	2	funeqi	$⊢ Fun (x \in A ⟼ B) \leftrightarrow Fun \{⟨x, y⟩ \| (x \in A \land y = B)\}$
4	1 3	mpbir	$⊢ Fun (x \in A ⟼ B)$