Theorem relttrcl 33508

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.

Step	Hyp	Ref	Expression
1		df-ttrcl 33504	. 2 ⊢ t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))}
2	1	relopabi 5689	1 ⊢ Rel t++𝑅

Syntax hints:   ∧ wa 399   ∧ w3a 1089   = wceq 1543  ∃wex 1787  ∀wral 3058  ∃wrex 3059   ∖ cdif 3860  ∅c0 4234   class class class wbr 5050  Rel wrel 5553  suc csuc 6212   Fn wfn 6372  ‘cfv 6377  ωcom 7641  1_oc1o 8192  t++cttrcl 33503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3067  df-v 3407  df-dif 3866  df-un 3868  df-in 3870  df-ss 3880  df-nul 4235  df-if 4437  df-sn 4539  df-pr 4541  df-op 4545  df-opab 5113  df-xp 5554  df-rel 5555  df-ttrcl 33504

This theorem is referenced by:  brttrcl  33509  ttrclss  33516  ttrclexg  33519