Theorem fnmpti

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Hypotheses	fnmpti.1	$$⊢B\in \mathrm{V}$$
	fnmpti.2	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	fnmpti	$$⊢FFnA$$

Proof

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