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Theorem relttrcl 9656
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 9652 . 2 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑚𝑛 (𝑓𝑚)𝑅(𝑓‘suc 𝑚))}
21relopabi 5782 1 Rel t++𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 397  w3a 1088   = wceq 1542  wex 1782  wral 3061  wrex 3070  cdif 3911  c0 4286   class class class wbr 5109  Rel wrel 5642  suc csuc 6323   Fn wfn 6495  cfv 6500  ωcom 7806  1oc1o 8409  t++cttrcl 9651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5172  df-xp 5643  df-rel 5644  df-ttrcl 9652
This theorem is referenced by:  brttrcl  9657  ttrclss  9664  ttrclexg  9667
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