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Theorem nfttrcld 33506
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 33504 . 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 1922 . . 3 𝑦𝜑
3 nfv 1922 . . 3 𝑧𝜑
4 nfv 1922 . . . 4 𝑛𝜑
5 nfcvd 2905 . . . 4 (𝜑𝑥(ω ∖ 1o))
6 nfv 1922 . . . . 5 𝑓𝜑
7 nfvd 1923 . . . . . 6 (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛)
8 nfvd 1923 . . . . . 6 (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧))
9 nfv 1922 . . . . . . 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 5096 . . . . . . 7 (𝜑 → Ⅎ𝑥(𝑓𝑎)𝑅(𝑓‘suc 𝑎))
159, 10, 14nfraldw 3141 . . . . . 6 (𝜑 → Ⅎ𝑥𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
167, 8, 15nf3and 1906 . . . . 5 (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
176, 16nfexd 2328 . . . 4 (𝜑 → Ⅎ𝑥𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
184, 5, 17nfrexd 3223 . . 3 (𝜑 → Ⅎ𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
192, 3, 18nfopabd 5118 . 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 399  w3a 1089   = wceq 1543  wex 1787  wnfc 2884  wral 3058  wrex 3059  cdif 3860  c0 4234   class class class wbr 5050  {copab 5112  suc csuc 6212   Fn wfn 6372  cfv 6377  ωcom 7641  1oc1o 8192  t++cttrcl 33503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3063  df-rex 3064  df-rab 3067  df-v 3407  df-dif 3866  df-un 3868  df-nul 4235  df-if 4437  df-sn 4539  df-pr 4541  df-op 4545  df-br 5051  df-opab 5113  df-ttrcl 33504
This theorem is referenced by:  nfttrcl  33507
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