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Theorem relttrcl 9750
Description: The transitive closure of a class is a relation. (Contributed by Scott Fenton, 17-Oct-2024.)
Assertion
Ref Expression
relttrcl Rel t++𝑅

Proof of Theorem relttrcl
Dummy variables 𝑓 𝑛 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ttrcl 9746 . 2 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚))}
21relopabi 5835 1 Rel t++𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1086   = wceq 1537  wex 1776  wral 3059  wrex 3068  cdif 3960  c0 4339   class class class wbr 5148  Rel wrel 5694  suc csuc 6388   Fn wfn 6558  cfv 6563  ωcom 7887  1oc1o 8498  t++cttrcl 9745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696  df-ttrcl 9746
This theorem is referenced by:  brttrcl  9751  ttrclss  9758  ttrclexg  9761
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