Theorem fnmpo

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

	Ref	Expression
Hypothesis	fmpo.1	$$⊢F=\left(x\in A,y\in B⟼C\right)$$
Assertion	fnmpo	$$⊢\forall x\in A\forall y\in BC\in V\to FFn\left(A\times B\right)$$

Proof

Step	Hyp	Ref	Expression
1		fmpo.1	$$⊢F=\left(x\in A,y\in B⟼C\right)$$
2		elex	$$⊢C\in V\to C\in \mathrm{V}$$
3	2	2ralimi	$$⊢\forall x\in A\forall y\in BC\in V\to \forall x\in A\forall y\in BC\in \mathrm{V}$$
4	1	fmpo	$$⊢\forall x\in A\forall y\in BC\in \mathrm{V}\leftrightarrow F:A\times B⟶\mathrm{V}$$
5		dffn2	$$⊢FFn\left(A\times B\right)\leftrightarrow F:A\times B⟶\mathrm{V}$$
6	4 5	bitr4i	$$⊢\forall x\in A\forall y\in BC\in \mathrm{V}\leftrightarrow FFn\left(A\times B\right)$$
7	3 6	sylib	$$⊢\forall x\in A\forall y\in BC\in V\to FFn\left(A\times B\right)$$