Theorem fnmptif

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by Glauco Siliprandi, 21-Dec-2024)

	Ref	Expression
Hypotheses	fnmptif.1	$$⊢\underset{\_}{Ⅎ}xA$$
	fnmptif.2	$$⊢B\in \mathrm{V}$$
	fnmptif.3	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	fnmptif	$$⊢FFnA$$

Proof

Step	Hyp	Ref	Expression
1		fnmptif.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		fnmptif.2	$$⊢B\in \mathrm{V}$$
3		fnmptif.3	$$⊢F=\left(x\in A⟼B\right)$$
4	2	rgenw	$$⊢\forall x\in AB\in \mathrm{V}$$
5	1	mptfnf	$$⊢\forall x\in AB\in \mathrm{V}\leftrightarrow \left(x\in A⟼B\right)FnA$$
6	4 5	mpbi	$$⊢\left(x\in A⟼B\right)FnA$$
7	3	fneq1i	$$⊢FFnA\leftrightarrow \left(x\in A⟼B\right)FnA$$
8	6 7	mpbir	$$⊢FFnA$$