Theorem relttrcl 9623

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 9619	. 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 5770	1 ⊢ Rel t++𝑅

Syntax hints:   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781  ∀wral 3050  ∃wrex 3059   ∖ cdif 3897  ∅c0 4284   class class class wbr 5097  Rel wrel 5628  suc csuc 6318   Fn wfn 6486  ‘cfv 6491  ωcom 7808  1_oc1o 8390  t++cttrcl 9618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2183  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5160  df-xp 5629  df-rel 5630  df-ttrcl 9619

This theorem is referenced by:  brttrcl  9624  ttrclss  9631  ttrclexg  9634