Theorem fnmpt

Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013)

	Ref	Expression
Hypothesis	mptfng.1	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	fnmpt	$$⊢\forall x\in AB\in V\to FFnA$$

Proof

Step	Hyp	Ref	Expression
1		mptfng.1	$$⊢F=\left(x\in A⟼B\right)$$
2		elex	$$⊢B\in V\to B\in \mathrm{V}$$
3	2	ralimi	$$⊢\forall x\in AB\in V\to \forall x\in AB\in \mathrm{V}$$
4	1	mptfng	$$⊢\forall x\in AB\in \mathrm{V}\leftrightarrow FFnA$$
5	3 4	sylib	$$⊢\forall x\in AB\in V\to FFnA$$