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Theorem ttrclresv 9758
Description: The transitive closure of 𝑅 restricted to V is the same as the transitive closure of 𝑅 itself. (Contributed by Scott Fenton, 20-Oct-2024.)
Assertion
Ref Expression
ttrclresv t++(𝑅 ↾ V) = t++𝑅

Proof of Theorem ttrclresv
Dummy variables 𝑓 𝑛 𝑎 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6918 . . . . . . . 8 (𝑓𝑎) ∈ V
2 fvex 6918 . . . . . . . . 9 (𝑓‘suc 𝑎) ∈ V
32brresi 6005 . . . . . . . 8 ((𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ((𝑓𝑎) ∈ V ∧ (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
41, 3mpbiran 709 . . . . . . 7 ((𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
54ralbii 3092 . . . . . 6 (∀𝑎𝑛 (𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
653anbi3i 1159 . . . . 5 ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎)) ↔ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
76exbii 1847 . . . 4 (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎)) ↔ ∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
87rexbii 3093 . . 3 (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎)) ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
98opabbii 5209 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
10 df-ttrcl 9749 . 2 t++(𝑅 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎))}
11 df-ttrcl 9749 . 2 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
129, 10, 113eqtr4i 2774 1 t++(𝑅 ↾ V) = t++𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  c0 4332   class class class wbr 5142  {copab 5204  cres 5686  suc csuc 6385   Fn wfn 6555  cfv 6560  ωcom 7888  1oc1o 8500  t++cttrcl 9748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-res 5696  df-iota 6513  df-fv 6568  df-ttrcl 9749
This theorem is referenced by:  ttrclco  9759  cottrcl  9760  dmttrcl  9762  rnttrcl  9763
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