Theorem relttrcl 9630

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 9626	. 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 5775	1 ⊢ Rel t++𝑅

Syntax hints:   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781  ∀wral 3052  ∃wrex 3062   ∖ cdif 3887  ∅c0 4274   class class class wbr 5086  Rel wrel 5633  suc csuc 6323   Fn wfn 6491  ‘cfv 6496  ωcom 7814  1_oc1o 8395  t++cttrcl 9625

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 2185  ax-ext 2709

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 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149  df-xp 5634  df-rel 5635  df-ttrcl 9626

This theorem is referenced by:  brttrcl  9631  ttrclss  9638  ttrclexg  9641