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	$$⊢F=\left(x\in A,y\in B⟼C\right)$$
Assertion	dmmpoga	$$⊢\forall x\in A\forall y\in BC\in V\to \mathrm{dom}F=A\times B$$

Proof

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