Theorem relttrcl 9737

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 9733	. 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 5805	1 ⊢ Rel t++𝑅

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812  ∀wral 3080  ∃wrex 3090   ∖ cdif 4048  ∅c0 4281   class class class wbr 5106  Rel wrel 5660  suc csuc 6379   Fn wfn 6550  ‘cfv 6555  ωcom 7898  1_oc1o 8494  t++cttrcl 9732

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-rel 5662  df-ttrcl 9733

This theorem is used by:  brttrcl  9738  ttrclss  9745  ttrclexg  9748