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Theorem ttrclresv 9661
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 6859 . . . . . . . 8 (𝑓𝑎) ∈ V
2 fvex 6859 . . . . . . . . 9 (𝑓‘suc 𝑎) ∈ V
32brresi 5950 . . . . . . . 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 3093 . . . . . 6 (∀𝑎𝑛 (𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
653anbi3i 1160 . . . . 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 1851 . . . 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 3094 . . 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 5176 . 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 9652 . 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 9652 . 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 2771 1 t++(𝑅 ↾ V) = t++𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wral 3061  wrex 3070  Vcvv 3447  cdif 3911  c0 4286   class class class wbr 5109  {copab 5171  cres 5639  suc csuc 6323   Fn wfn 6495  cfv 6500  ωcom 7806  1oc1o 8409  t++cttrcl 9651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-res 5649  df-iota 6452  df-fv 6508  df-ttrcl 9652
This theorem is referenced by:  ttrclco  9662  cottrcl  9663  dmttrcl  9665  rnttrcl  9666
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