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Theorem nfofr 5956
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 5951 . 2 𝑟 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
2 nfcv 2258 . . . 4 𝑥(dom 𝑢 ∩ dom 𝑣)
3 nfcv 2258 . . . . 5 𝑥(𝑢𝑤)
4 nfof.1 . . . . 5 𝑥𝑅
5 nfcv 2258 . . . . 5 𝑥(𝑣𝑤)
63, 4, 5nfbr 3944 . . . 4 𝑥(𝑢𝑤)𝑅(𝑣𝑤)
72, 6nfralxy 2448 . . 3 𝑥𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)
87nfopab 3966 . 2 𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
91, 8nfcxfr 2255 1 𝑥𝑟 𝑅
Colors of variables: wff set class
Syntax hints:  wnfc 2245  wral 2393  cin 3040   class class class wbr 3899  {copab 3958  dom cdm 4509  cfv 5093  𝑟 cofr 5949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-ofr 5951
This theorem is referenced by: (None)
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