Theorem tmachlem-uassst 47756

Description: Union of all agreement sets only includes tapes. (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-uassst	⊢ (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-uassst

Step	Hyp	Ref	Expression
1		tmach.agreemap	. . . . 5 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
2	1	rneqd 5926	. . . 4 ⊢ (𝜑 → ran 𝐴 = ran (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
3	2	unieqd 4883	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 = ∪ ran (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
4		tmach.finalph	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Fin)
5		tmach.exindex	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
6		tmach.tapelist	. . . . . . . 8 ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
7		tmach.scanmap	. . . . . . . 8 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
8		tmach.agreement	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))
9	4, 5, 6, 7, 1, 8	tmachlem-extapes 47747	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ V)
10	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑇) → 𝑇 ∈ V)
11		ssrab2 4031	. . . . . . 7 ⊢ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ⊆ 𝑇
12	11	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑇) → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ⊆ 𝑇)
13	10, 12	ssexd 5293	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑇) → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
14	13	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
15		dfiun3g 5956	. . . 4 ⊢ (∀𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V → ∪ 𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} = ∪ ran (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
16	14, 15	syl 18	. . 3 ⊢ (𝜑 → ∪ 𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} = ∪ ran (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
17	3, 16	eqtr4d 2800	. 2 ⊢ (𝜑 → ∪ ran 𝐴 = ∪ 𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))})
18	12	iunssd 5013	. 2 ⊢ (𝜑 → ∪ 𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ⊆ 𝑇)
19	17, 18	eqsstrd 3968	1 ⊢ (𝜑 → ∪ ran 𝐴 ⊆ 𝑇)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5190  ran crn 5660   ↾ 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-11 2194  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-nfc 2911  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-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-cnv 5667  df-dm 5669  df-rn 5670  df-iota 6493  df-fv 6545  df-ov 7419

This theorem is used by:  tmachlem-exlargecover  47757