Theorem tmachlem-extpcover 48158

Description: Product topology of tapes admits finite 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-extpcover	⊢ (𝜑 → ∃𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin)∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)

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

Proof of Theorem tmachlem-extpcover

Dummy variable 𝑏 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-tpcomp 48151	. 2 ⊢ (𝜑 → (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) ∈ Comp)
8	1, 2, 3, 4, 5, 6	tmachlem-extapes 48147	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ V)
9		ssrab2 4008	. . . . . . . 8 ⊢ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ⊆ 𝑇
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ⊆ 𝑇)
11	8, 10	ssexd 5290	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
12	11	ralrimivw 3162	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
13		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑧𝑇
14	13	mptfnf 6693	. . . . 5 ⊢ (∀𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V ↔ (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}) Fn 𝑇)
15	12, 14	sylib 221	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}) Fn 𝑇)
16	5	fneq1d 6651	. . . 4 ⊢ (𝜑 → (𝐴 Fn 𝑇 ↔ (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}) Fn 𝑇))
17	15, 16	mpbird 260	. . 3 ⊢ (𝜑 → 𝐴 Fn 𝑇)
18	1, 2, 3, 4, 5, 6	tmachlem-tpopen2 48155	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑇) → (𝐴‘𝑏) ∈ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)))
19	18	ralrimiva 3158	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝑇 (𝐴‘𝑏) ∈ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)))
20		fnfvrnss 7142	. . 3 ⊢ ((𝐴 Fn 𝑇 ∧ ∀𝑏 ∈ 𝑇 (𝐴‘𝑏) ∈ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈))) → ran 𝐴 ⊆ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)))
21	17, 19, 20	syl2anc 596	. 2 ⊢ (𝜑 → ran 𝐴 ⊆ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)))
22	1, 2, 3, 4, 5, 6	tmachlem-tpbase 48152	. . 3 ⊢ (𝜑 → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = 𝑇)
23	1, 2, 3, 4, 5, 6	tmachlem-exlargecover 48157	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 = 𝑇)
24	22, 23	eqtr4d 2801	. 2 ⊢ (𝜑 → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ ran 𝐴)
25		eqid 2763	. . 3 ⊢ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈))
26	25	cmpcov 23734	. 2 ⊢ (((∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) ∈ Comp ∧ ran 𝐴 ⊆ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ ran 𝐴) → ∃𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin)∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)
27	7, 21, 24, 26	syl3anc 1399	1 ⊢ (𝜑 → ∃𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin)∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4870   ↦ cmpt 5189  ran crn 5656   ↾ cres 5657   Fn wfn 6550  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872  Fincfn 8998  ∏_tcpt 17635  Compccmp 23731

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-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-ac2 10565

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-rpss 7760  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-omul 8506  df-er 8742  df-map 8874  df-ixp 8951  df-en 8999  df-dom 9000  df-fin 9002  df-fi 9427  df-wdom 9583  df-dju 10006  df-card 10044  df-acn 10047  df-ac 10219  df-topgen 17640  df-pt 17641  df-fbas 21702  df-fg 21703  df-top 23239  df-topon 23256  df-bases 23291  df-cld 23364  df-ntr 23365  df-cls 23366  df-nei 23443  df-cmp 23732  df-fil 24192  df-ufil 24247  df-ufl 24248  df-flim 24285  df-fcls 24287

This theorem is used by:  tmachlem-exagreecover  48159