Theorem nfofr 7660

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

Hypothesis

Ref	Expression
nfof.1	⊢ Ⅎ𝑥𝑅

Assertion

Ref	Expression
nfofr	⊢ Ⅎ𝑥 ∘r 𝑅

Proof of Theorem nfofr

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ofr 7654	. 2 ⊢ ∘r 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
2		nfcv 2891	. . . 4 ⊢ Ⅎ𝑥(dom 𝑢 ∩ dom 𝑣)
3		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥(𝑢‘𝑤)
4		nfof.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
5		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥(𝑣‘𝑤)
6	3, 4, 5	nfbr 5154	. . . 4 ⊢ Ⅎ𝑥(𝑢‘𝑤)𝑅(𝑣‘𝑤)
7	2, 6	nfralw 3285	. . 3 ⊢ Ⅎ𝑥∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)
8	7	nfopab 5176	. 2 ⊢ Ⅎ𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢‘𝑤)𝑅(𝑣‘𝑤)}
9	1, 8	nfcxfr 2889	1 ⊢ Ⅎ𝑥 ∘r 𝑅

Syntax hints:  Ⅎwnfc 2876  ∀wral 3044   ∩ cin 3913   class class class wbr 5107  {copab 5169  dom cdm 5638  ‘cfv 6511   ∘_r cofr 7652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-ofr 7654

This theorem is referenced by: (None)