Theorem nffun

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

	Ref	Expression
Hypothesis	nffun.1	$$⊢\underset{\_}{Ⅎ}xF$$
Assertion	nffun	$$⊢Ⅎx\mathrm{Fun}F$$

Proof

Step	Hyp	Ref	Expression
1		nffun.1	$$⊢\underset{\_}{Ⅎ}xF$$
2		df-fun	$$⊢\mathrm{Fun}F\leftrightarrow \left(\mathrm{Rel}F\wedge F\circ F^{-1}\subseteq \mathrm{I}\right)$$
3	1	nfrel	$$⊢Ⅎx\mathrm{Rel}F$$
4	1	nfcnv	$$⊢\underset{\_}{Ⅎ}x{F}^{-1}$$
5	1 4	nfco	$$⊢\underset{\_}{Ⅎ}x\left(F\circ F^{-1}\right)$$
6		nfcv	$$⊢\underset{\_}{Ⅎ}x\mathrm{I}$$
7	5 6	nfss	$$⊢ℲxF\circ F^{-1}\subseteq \mathrm{I}$$
8	3 7	nfan	$$⊢Ⅎx\left(\mathrm{Rel}F\wedge F\circ F^{-1}\subseteq \mathrm{I}\right)$$
9	2 8	nfxfr	$$⊢Ⅎx\mathrm{Fun}F$$