Theorem nfof

Description: Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014)

	Ref	Expression
Hypothesis	nfof.1	$$⊢\underset{\_}{Ⅎ}xR$$
Assertion	nfof	$$⊢\underset{\_}{Ⅎ}x{\circ }_{f}R$$

Proof

Step	Hyp	Ref	Expression
1		nfof.1	$$⊢\underset{\_}{Ⅎ}xR$$
2		df-of	$$⊢{\circ }_{f}R=\left(u\in \mathrm{V},v\in \mathrm{V}⟼\left(w\in \left(\mathrm{dom}u\cap \mathrm{dom}v\right)⟼u\left(w\right)Rv\left(w\right)\right)\right)$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}x\mathrm{V}$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}x\left(\mathrm{dom}u\cap \mathrm{dom}v\right)$$
5		nfcv	$$⊢\underset{\_}{Ⅎ}xu\left(w\right)$$
6		nfcv	$$⊢\underset{\_}{Ⅎ}xv\left(w\right)$$
7	5 1 6	nfov	$$⊢\underset{\_}{Ⅎ}x\left(u\left(w\right)Rv\left(w\right)\right)$$
8	4 7	nfmpt	$$⊢\underset{\_}{Ⅎ}x\left(w\in \left(\mathrm{dom}u\cap \mathrm{dom}v\right)⟼u\left(w\right)Rv\left(w\right)\right)$$
9	3 3 8	nfmpo	$$⊢\underset{\_}{Ⅎ}x\left(u\in \mathrm{V},v\in \mathrm{V}⟼\left(w\in \left(\mathrm{dom}u\cap \mathrm{dom}v\right)⟼u\left(w\right)Rv\left(w\right)\right)\right)$$
10	2 9	nfcxfr	$$⊢\underset{\_}{Ⅎ}x{\circ }_{f}R$$