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Theorem nfofr 7671
Description: Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypothesis
Ref Expression
nfof.1 𝑥𝑅
Assertion
Ref Expression
nfofr 𝑥r 𝑅

Proof of Theorem nfofr
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ofr 7665 . 2 r 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
2 nfcv 2895 . . . 4 𝑥(dom 𝑢 ∩ dom 𝑣)
3 nfcv 2895 . . . . 5 𝑥(𝑢𝑤)
4 nfof.1 . . . . 5 𝑥𝑅
5 nfcv 2895 . . . . 5 𝑥(𝑣𝑤)
63, 4, 5nfbr 5186 . . . 4 𝑥(𝑢𝑤)𝑅(𝑣𝑤)
72, 6nfralw 3300 . . 3 𝑥𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)
87nfopab 5208 . 2 𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
91, 8nfcxfr 2893 1 𝑥r 𝑅
Colors of variables: wff setvar class
Syntax hints:  wnfc 2875  wral 3053  cin 3940   class class class wbr 5139  {copab 5201  dom cdm 5667  cfv 6534  r cofr 7663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-ofr 7665
This theorem is referenced by: (None)
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