Theorem fnmptf

Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013) (Revised by Thierry Arnoux, 10-May-2017)

	Ref	Expression
Hypothesis	mptfnf.0	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	fnmptf	$$⊢\forall x\in AB\in V\to \left(x\in A⟼B\right)FnA$$

Proof

Step	Hyp	Ref	Expression
1		mptfnf.0	$$⊢\underset{\_}{Ⅎ}xA$$
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	mptfnf	$$⊢\forall x\in AB\in \mathrm{V}\leftrightarrow \left(x\in A⟼B\right)FnA$$
5	3 4	sylib	$$⊢\forall x\in AB\in V\to \left(x\in A⟼B\right)FnA$$