Theorem fnmptd

Description: The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	fnmptd.1	$$⊢Ⅎx\phi$$
	fnmptd.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
	fnmptd.3	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	fnmptd	$$⊢\phi \to FFnA$$

Proof

Step	Hyp	Ref	Expression
1		fnmptd.1	$$⊢Ⅎx\phi$$
2		fnmptd.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
3		fnmptd.3	$$⊢F=\left(x\in A⟼B\right)$$
4	2	ex	$$⊢\phi \to \left(x\in A\to B\in V\right)$$
5	1 4	ralrimi	$$⊢\phi \to \forall x\in AB\in V$$
6	3	fnmpt	$$⊢\forall x\in AB\in V\to FFnA$$
7	5 6	syl	$$⊢\phi \to FFnA$$