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

Theorem nfttrcld 9654
Description: Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
Hypothesis
Ref Expression
nfttrcld.1 (𝜑𝑥𝑅)
Assertion
Ref Expression
nfttrcld (𝜑𝑥t++𝑅)

Proof of Theorem nfttrcld
Dummy variables 𝑦 𝑧 𝑛 𝑓 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ttrcl 9652 . 2 t++𝑅 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
2 nfv 1918 . . 3 𝑦𝜑
3 nfv 1918 . . 3 𝑧𝜑
4 nfv 1918 . . . 4 𝑛𝜑
5 nfcvd 2905 . . . 4 (𝜑𝑥(ω ∖ 1o))
6 nfv 1918 . . . . 5 𝑓𝜑
7 nfvd 1919 . . . . . 6 (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛)
8 nfvd 1919 . . . . . 6 (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧))
9 nfv 1918 . . . . . . 7 𝑎𝜑
10 nfcvd 2905 . . . . . . 7 (𝜑𝑥𝑛)
11 nfcvd 2905 . . . . . . . 8 (𝜑𝑥(𝑓𝑎))
12 nfttrcld.1 . . . . . . . 8 (𝜑𝑥𝑅)
13 nfcvd 2905 . . . . . . . 8 (𝜑𝑥(𝑓‘suc 𝑎))
1411, 12, 13nfbrd 5155 . . . . . . 7 (𝜑 → Ⅎ𝑥(𝑓𝑎)𝑅(𝑓‘suc 𝑎))
159, 10, 14nfraldw 3291 . . . . . 6 (𝜑 → Ⅎ𝑥𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
167, 8, 15nf3and 1902 . . . . 5 (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
176, 16nfexd 2323 . . . 4 (𝜑 → Ⅎ𝑥𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
184, 5, 17nfrexdw 3292 . . 3 (𝜑 → Ⅎ𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
192, 3, 18nfopabd 5177 . 2 (𝜑𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))})
201, 19nfcxfrd 2903 1 (𝜑𝑥t++𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wex 1782  wnfc 2884  wral 3061  wrex 3070  cdif 3911  c0 4286   class class class wbr 5109  {copab 5171  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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
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-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  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-br 5110  df-opab 5172  df-ttrcl 9652
This theorem is referenced by:  nfttrcl  9655
  Copyright terms: Public domain W3C validator