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Theorem relttrcl 9753
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 9749 . 2 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚))}
21relopabi 5831 1 Rel t++𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1086   = wceq 1539  wex 1778  wral 3060  wrex 3069  cdif 3947  c0 4332   class class class wbr 5142  Rel wrel 5689  suc csuc 6385   Fn wfn 6555  cfv 6560  ωcom 7888  1oc1o 8500  t++cttrcl 9748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-xp 5690  df-rel 5691  df-ttrcl 9749
This theorem is referenced by:  brttrcl  9754  ttrclss  9761  ttrclexg  9764
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