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Theorem nfofr 6103
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 6098 . 2 𝑟 𝑅 = {⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
2 nfcv 2329 . . . 4 𝑥(dom 𝑢 ∩ dom 𝑣)
3 nfcv 2329 . . . . 5 𝑥(𝑢𝑤)
4 nfof.1 . . . . 5 𝑥𝑅
5 nfcv 2329 . . . . 5 𝑥(𝑣𝑤)
63, 4, 5nfbr 4061 . . . 4 𝑥(𝑢𝑤)𝑅(𝑣𝑤)
72, 6nfralxy 2525 . . 3 𝑥𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)
87nfopab 4083 . 2 𝑥{⟨𝑢, 𝑣⟩ ∣ ∀𝑤 ∈ (dom 𝑢 ∩ dom 𝑣)(𝑢𝑤)𝑅(𝑣𝑤)}
91, 8nfcxfr 2326 1 𝑥𝑟 𝑅
Colors of variables: wff set class
Syntax hints:  wnfc 2316  wral 2465  cin 3140   class class class wbr 4015  {copab 4075  dom cdm 4638  cfv 5228  𝑟 cofr 6096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-ofr 6098
This theorem is referenced by: (None)
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