Theorem tmachlem-extapes 47747

Description: The class of all tapes is a set. (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-extapes	⊢ (𝜑 → 𝑇 ∈ V)

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

Proof of Theorem tmachlem-extapes

Step	Hyp	Ref	Expression
1		tmach.tapelist	. 2 ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
2		ovex 7449	. 2 ⊢ (𝑈 ↑m 𝐼) ∈ V
3	1, 2	eqeltrdi 2870	1 ⊢ (𝜑 → 𝑇 ∈ V)

This proof depends on syntax axioms:   → wi 4   = 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-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-sn 4588  df-pr 4590  df-uni 4871  df-iota 6493  df-fv 6545  df-ov 7419

This theorem is used by:  tmachlem-agreeself  47749  tmachlem-agreeprod  47750  tmachlem-uassst  47756  tmachlem-extpcover  47758  tmachlem-agreefin  47761