Theorem nffn

Description: Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004)

	Ref	Expression
Hypotheses	nffn.1	$$⊢\underset{\_}{Ⅎ}xF$$
	nffn.2	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	nffn	$$⊢ℲxFFnA$$

Proof

Step	Hyp	Ref	Expression
1		nffn.1	$$⊢\underset{\_}{Ⅎ}xF$$
2		nffn.2	$$⊢\underset{\_}{Ⅎ}xA$$
3		df-fn	$$⊢FFnA\leftrightarrow \left(\mathrm{Fun}F\wedge \mathrm{dom}F=A\right)$$
4	1	nffun	$$⊢Ⅎx\mathrm{Fun}F$$
5	1	nfdm	$$⊢\underset{\_}{Ⅎ}x\mathrm{dom}F$$
6	5 2	nfeq	$$⊢Ⅎx\mathrm{dom}F=A$$
7	4 6	nfan	$$⊢Ⅎx\left(\mathrm{Fun}F\wedge \mathrm{dom}F=A\right)$$
8	3 7	nfxfr	$$⊢ℲxFFnA$$