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Theorem nfttrcld 9748
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 9746 . 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 1912 . . 3 𝑦𝜑
3 nfv 1912 . . 3 𝑧𝜑
4 nfv 1912 . . . 4 𝑛𝜑
5 nfcvd 2904 . . . 4 (𝜑𝑥(ω ∖ 1o))
6 nfv 1912 . . . . 5 𝑓𝜑
7 nfvd 1913 . . . . . 6 (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛)
8 nfvd 1913 . . . . . 6 (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧))
9 nfv 1912 . . . . . . 7 𝑎𝜑
10 nfcvd 2904 . . . . . . 7 (𝜑𝑥𝑛)
11 nfcvd 2904 . . . . . . . 8 (𝜑𝑥(𝑓𝑎))
12 nfttrcld.1 . . . . . . . 8 (𝜑𝑥𝑅)
13 nfcvd 2904 . . . . . . . 8 (𝜑𝑥(𝑓‘suc 𝑎))
1411, 12, 13nfbrd 5194 . . . . . . 7 (𝜑 → Ⅎ𝑥(𝑓𝑎)𝑅(𝑓‘suc 𝑎))
159, 10, 14nfraldw 3307 . . . . . 6 (𝜑 → Ⅎ𝑥𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
167, 8, 15nf3and 1896 . . . . 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, 17nfrexdw 3308 . . 3 (𝜑 → Ⅎ𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
192, 3, 18nfopabd 5216 . 2 (𝜑𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))})
201, 19nfcxfrd 2902 1 (𝜑𝑥t++𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wnfc 2888  wral 3059  wrex 3068  cdif 3960  c0 4339   class class class wbr 5148  {copab 5210  suc csuc 6388   Fn wfn 6558  cfv 6563  ωcom 7887  1oc1o 8498  t++cttrcl 9745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-ttrcl 9746
This theorem is referenced by:  nfttrcl  9749
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