Theorem tmachlem-exlargecover 47954

Description: Product topology of tapes has an open cover. (Contributed by Ender Ting, 28-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-exlargecover	⊢ (𝜑 → ∪ ran 𝐴 = 𝑇)

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-exlargecover

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tmach.finalph	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
2		tmach.exindex	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
3		tmach.tapelist	. . 3 ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
4		tmach.scanmap	. . 3 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
5		tmach.agreemap	. . 3 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
6		tmach.agreement	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))
7	1, 2, 3, 4, 5, 6	tmachlem-uassst 47953	. 2 ⊢ (𝜑 → ∪ ran 𝐴 ⊆ 𝑇)
8	1, 2, 3, 4, 5, 6	tmachlem-agreeself 47946	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝐴‘𝑎))
9		elfvunirn 6915	. . 3 ⊢ (𝑎 ∈ (𝐴‘𝑎) → 𝑎 ∈ ∪ ran 𝐴)
10	8, 9	syl 18	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ ∪ ran 𝐴)
11	7, 10	eqelssd 3929	1 ⊢ (𝜑 → ∪ ran 𝐴 = 𝑇)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ∩ cin 4089  𝒫 cpw 4556  ∪ cuni 4866   ↦ cmpt 5185  ran crn 5651   ↾ cres 5652  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  Fincfn 8973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423

This theorem is used by:  tmachlem-extpcover  47955