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Theorem ttrclresv 9741
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 6910 . . . . . . . 8 (𝑓𝑎) ∈ V
2 fvex 6910 . . . . . . . . 9 (𝑓‘suc 𝑎) ∈ V
32brresi 5994 . . . . . . . 8 ((𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ((𝑓𝑎) ∈ V ∧ (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
41, 3mpbiran 708 . . . . . . 7 ((𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
54ralbii 3090 . . . . . 6 (∀𝑎𝑛 (𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
653anbi3i 1157 . . . . 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 1843 . . . 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 3091 . . 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 5215 . 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 9732 . 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 9732 . 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 2766 1 t++(𝑅 ↾ V) = t++𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  wral 3058  wrex 3067  Vcvv 3471  cdif 3944  c0 4323   class class class wbr 5148  {copab 5210  cres 5680  suc csuc 6371   Fn wfn 6543  cfv 6548  ωcom 7870  1oc1o 8480  t++cttrcl 9731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-res 5690  df-iota 6500  df-fv 6556  df-ttrcl 9732
This theorem is referenced by:  ttrclco  9742  cottrcl  9743  dmttrcl  9745  rnttrcl  9746
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