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Theorem nfttrcld 9741
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 9739 . 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 1909 . . 3 𝑦𝜑
3 nfv 1909 . . 3 𝑧𝜑
4 nfv 1909 . . . 4 𝑛𝜑
5 nfcvd 2900 . . . 4 (𝜑𝑥(ω ∖ 1o))
6 nfv 1909 . . . . 5 𝑓𝜑
7 nfvd 1910 . . . . . 6 (𝜑 → Ⅎ𝑥 𝑓 Fn suc 𝑛)
8 nfvd 1910 . . . . . 6 (𝜑 → Ⅎ𝑥((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧))
9 nfv 1909 . . . . . . 7 𝑎𝜑
10 nfcvd 2900 . . . . . . 7 (𝜑𝑥𝑛)
11 nfcvd 2900 . . . . . . . 8 (𝜑𝑥(𝑓𝑎))
12 nfttrcld.1 . . . . . . . 8 (𝜑𝑥𝑅)
13 nfcvd 2900 . . . . . . . 8 (𝜑𝑥(𝑓‘suc 𝑎))
1411, 12, 13nfbrd 5198 . . . . . . 7 (𝜑 → Ⅎ𝑥(𝑓𝑎)𝑅(𝑓‘suc 𝑎))
159, 10, 14nfraldw 3304 . . . . . 6 (𝜑 → Ⅎ𝑥𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
167, 8, 15nf3and 1893 . . . . 5 (𝜑 → Ⅎ𝑥(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
176, 16nfexd 2317 . . . 4 (𝜑 → Ⅎ𝑥𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
184, 5, 17nfrexdw 3305 . . 3 (𝜑 → Ⅎ𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)))
192, 3, 18nfopabd 5220 . 2 (𝜑𝑥{⟨𝑦, 𝑧⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓𝑛) = 𝑧) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))})
201, 19nfcxfrd 2898 1 (𝜑𝑥t++𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wex 1773  wnfc 2879  wral 3058  wrex 3067  cdif 3946  c0 4326   class class class wbr 5152  {copab 5214  suc csuc 6376   Fn wfn 6548  cfv 6553  ωcom 7876  1oc1o 8486  t++cttrcl 9738
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699
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-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  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-br 5153  df-opab 5215  df-ttrcl 9739
This theorem is referenced by:  nfttrcl  9742
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