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Metamath Proof Explorer

Theorem dmmpoga

Description: Domain of an operation given by the maps-to notation, closed form of dmmpo . (Contributed by Alexander van der Vekens, 10-Feb-2019) (Proof shortened by Lammen, 29-May-2024)

		Ref	Expression
	Hypothesis	dmmpog.f	$⊢ F = (x \in A, y \in B ⟼ C)$
	Assertion	dmmpoga	$⊢ \forall x \in A \forall y \in B C \in V \to dom F = A \times B$

Proof

Step	Hyp	Ref	Expression
1		dmmpog.f	$⊢ F = (x \in A, y \in B ⟼ C)$
2	1	fnmpo	$⊢ \forall x \in A \forall y \in B C \in V \to F Fn (A \times B)$
3	2	fndmd	$⊢ \forall x \in A \forall y \in B C \in V \to dom F = A \times B$