Theorem dmmpo

Description: Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010)

	Ref	Expression
Hypotheses	fmpo.1	$$⊢F=\left(x\in A,y\in B⟼C\right)$$
	fnmpoi.2	$$⊢C\in \mathrm{V}$$
Assertion	dmmpo	$$⊢\mathrm{dom}F=A\times B$$

Proof

Step	Hyp	Ref	Expression
1		fmpo.1	$$⊢F=\left(x\in A,y\in B⟼C\right)$$
2		fnmpoi.2	$$⊢C\in \mathrm{V}$$
3	1 2	fnmpoi	$$⊢FFn\left(A\times B\right)$$
4		fndm	$$⊢FFn\left(A\times B\right)\to \mathrm{dom}F=A\times B$$
5	3 4	ax-mp	$$⊢\mathrm{dom}F=A\times B$$