Theorem tmachlem-finscan 47748

Description: Execution on any tape only scanned a finite number of cells. (Contributed by Ender Ting, 27-Jul-2026.)

Hypotheses

Ref	Expression
tmach.finalph	⊢ (𝜑 → 𝑈 ∈ Fin)
tmach.exindex	⊢ (𝜑 → 𝐼 ∈ V)
tmach.tapelist	⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
tmach.scanmap	⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
tmach.agreemap	⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
tmach.agreement	⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))

Assertion

Ref	Expression
tmachlem-finscan	⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆‘𝑎) ∈ Fin)

Distinct variable groups:   𝑈,𝑎,𝑦,𝑧   𝐼,𝑎,𝑦,𝑧   𝜑,𝑎,𝑦,𝑧   𝑆,𝑎,𝑦,𝑧   𝐴,𝑎,𝑦,𝑧   𝑇,𝑎,𝑦,𝑧

Proof of Theorem tmachlem-finscan

Step	Hyp	Ref	Expression
1		tmach.scanmap	. . 3 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
2	1	ffvelcdmda 7080	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆‘𝑎) ∈ (𝒫 𝐼 ∩ Fin))
3	2	elin2d 4154	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆‘𝑎) ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453   ∩ cin 3901  𝒫 cpw 4560   ↦ cmpt 5190   ↾ cres 5661  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ↑_m cmap 8829  Fincfn 8955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is used by:  tmachlem-tpopen  47754