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Theorem ttrclresv 9673
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 6881 . . . . . . . 8 (𝑓𝑎) ∈ V
2 fvex 6881 . . . . . . . . 9 (𝑓‘suc 𝑎) ∈ V
32brresi 5975 . . . . . . . 8 ((𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ((𝑓𝑎) ∈ V ∧ (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
41, 3mpbiran 719 . . . . . . 7 ((𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
54ralbii 3109 . . . . . 6 (∀𝑎𝑛 (𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
653anbi3i 1173 . . . . 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 1869 . . . 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 3110 . . 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 5168 . 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 9664 . 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 9664 . 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 2796 1 t++(𝑅 ↾ V) = t++𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 399  w3a 1099   = wceq 1561  wex 1800  wcel 2143  wral 3077  wrex 3087  Vcvv 3455  cdif 3902  c0 4286   class class class wbr 5101  {copab 5163  cres 5650  suc csuc 6349   Fn wfn 6517  cfv 6522  ωcom 7847  1oc1o 8431  t++cttrcl 9663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-xp 5654  df-res 5660  df-iota 6478  df-fv 6530  df-ttrcl 9664
This theorem is referenced by:  ttrclco  9674  cottrcl  9675  dmttrcl  9677  rnttrcl  9678
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