MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ttrclresv Structured version   Visualization version   GIF version

Theorem ttrclresv 9750
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 6915 . . . . . . . 8 (𝑓𝑎) ∈ V
2 fvex 6915 . . . . . . . . 9 (𝑓‘suc 𝑎) ∈ V
32brresi 5998 . . . . . . . 8 ((𝑓𝑎)(𝑅 ↾ V)(𝑓‘suc 𝑎) ↔ ((𝑓𝑎) ∈ V ∧ (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
41, 3mpbiran 707 . . . . . . 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 1156 . . . . 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 1842 . . . 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 5219 . 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 9741 . 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 9741 . 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 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wral 3058  wrex 3067  Vcvv 3473  cdif 3946  c0 4326   class class class wbr 5152  {copab 5214  cres 5684  suc csuc 6376   Fn wfn 6548  cfv 6553  ωcom 7878  1oc1o 8488  t++cttrcl 9740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-res 5694  df-iota 6505  df-fv 6561  df-ttrcl 9741
This theorem is referenced by:  ttrclco  9751  cottrcl  9752  dmttrcl  9754  rnttrcl  9755
  Copyright terms: Public domain W3C validator