Theorem tmachlem-uassst 48156

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 48147	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ V)
10	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑇) → 𝑇 ∈ V)
11		ssrab2 4008	. . . . . . 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 5290	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑇) → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
14	13	ralrimiva 3158	. . . 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 2801	. 2 ⊢ (𝜑 → ∪ ran 𝐴 = ∪ 𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))})
18	12	iunssd 5012	. 2 ⊢ (𝜑 → ∪ 𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ⊆ 𝑇)
19	17, 18	eqsstrd 3947	1 ⊢ (𝜑 → ∪ ran 𝐴 ⊆ 𝑇)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5189  ran crn 5656   ↾ cres 5657  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872  Fincfn 8998

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-cnv 5663  df-dm 5665  df-rn 5666  df-iota 6509  df-fv 6563  df-ov 7443

This theorem is used by:  tmachlem-exlargecover  48157