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Metamath Proof Explorer

Theorem fvmptndm

Description: Value of a function given by the maps-to notation, outside of its domain. (Contributed by AV, 31-Dec-2020)

		Ref	Expression
	Hypothesis	fvmptndm.1	$⊢ F = (x \in A ⟼ B)$
	Assertion	fvmptndm	$⊢ \neg X \in A \to F (X) = \emptyset$

Step	Hyp	Ref	Expression
1		fvmptndm.1	$⊢ F = (x \in A ⟼ B)$
2		df-mpt	$⊢ (x \in A ⟼ B) = \{⟨x, y⟩ \| (x \in A \land y = B)\}$
3	1 2	eqtri	$⊢ F = \{⟨x, y⟩ \| (x \in A \land y = B)\}$
4	3	fvopab4ndm	$⊢ \neg X \in A \to F (X) = \emptyset$