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Theorem nfofr 5996
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 5991 . 2 𝑟 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
2 nfcv 2282 . . . 4 𝑥(dom 𝑢 ∩ dom 𝑣)
3 nfcv 2282 . . . . 5 𝑥(𝑢𝑤)
4 nfof.1 . . . . 5 𝑥𝑅
5 nfcv 2282 . . . . 5 𝑥(𝑣𝑤)
63, 4, 5nfbr 3982 . . . 4 𝑥(𝑢𝑤)𝑅(𝑣𝑤)
72, 6nfralxy 2474 . . 3 𝑥𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)
87nfopab 4004 . 2 𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
91, 8nfcxfr 2279 1 𝑥𝑟 𝑅
Colors of variables: wff set class
Syntax hints:  wnfc 2269  wral 2417  cin 3075   class class class wbr 3937  {copab 3996  dom cdm 4547  cfv 5131  𝑟 cofr 5989
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-ofr 5991
This theorem is referenced by: (None)
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