Theorem nfofr

Description: Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014)

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

Proof

Step	Hyp	Ref	Expression
1		nfof.1	$$⊢\underset{\_}{Ⅎ}xR$$
2		df-ofr	$$⊢{\circ }_{r}R=\left\{⟨u,v⟩|\forall w\in \left(\mathrm{dom}u\cap \mathrm{dom}v\right)u\left(w\right)Rv\left(w\right)\right\}$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}x\left(\mathrm{dom}u\cap \mathrm{dom}v\right)$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}xu\left(w\right)$$
5		nfcv	$$⊢\underset{\_}{Ⅎ}xv\left(w\right)$$
6	4 1 5	nfbr	$$⊢Ⅎxu\left(w\right)Rv\left(w\right)$$
7	3 6	nfralw	$$⊢Ⅎx\forall w\in \left(\mathrm{dom}u\cap \mathrm{dom}v\right)u\left(w\right)Rv\left(w\right)$$
8	7	nfopab	$$⊢\underset{\_}{Ⅎ}x\left\{⟨u,v⟩|\forall w\in \left(\mathrm{dom}u\cap \mathrm{dom}v\right)u\left(w\right)Rv\left(w\right)\right\}$$
9	2 8	nfcxfr	$$⊢\underset{\_}{Ⅎ}x{\circ }_{r}R$$