Theorem tmachlem-extpcover 47838

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 47831	. 2 ⊢ (𝜑 → (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) ∈ Comp)
8	1, 2, 3, 4, 5, 6	tmachlem-extapes 47827	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ V)
9		ssrab2 4028	. . . . . . . 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 5289	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
12	11	ralrimivw 3158	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
13		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑧𝑇
14	13	mptfnf 6670	. . . . 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 6628	. . . 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 47835	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑇) → (𝐴‘𝑏) ∈ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)))
19	18	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝑇 (𝐴‘𝑏) ∈ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)))
20		fnfvrnss 7117	. . 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 47832	. . 3 ⊢ (𝜑 → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = 𝑇)
23	1, 2, 3, 4, 5, 6	tmachlem-exlargecover 47837	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 = 𝑇)
24	22, 23	eqtr4d 2798	. 2 ⊢ (𝜑 → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ ran 𝐴)
25		eqid 2760	. . 3 ⊢ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈))
26	25	cmpcov 23646	. 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 1398	1 ⊢ (𝜑 → ∃𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin)∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   ↦ cmpt 5186  ran crn 5656   ↾ cres 5657   Fn wfn 6530  ⟶wf 6531  ‘cfv 6535  (class class class)co 7416   ↑_m cmap 8833  Fincfn 8959  ∏_tcpt 17548  Compccmp 23643

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 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-ac2 10490

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-rpss 7730  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-er 8703  df-map 8835  df-ixp 8912  df-en 8960  df-dom 8961  df-fin 8963  df-fi 9388  df-wdom 9544  df-dju 9931  df-card 9969  df-acn 9972  df-ac 10144  df-topgen 17553  df-pt 17554  df-fbas 21614  df-fg 21615  df-top 23151  df-topon 23168  df-bases 23203  df-cld 23276  df-ntr 23277  df-cls 23278  df-nei 23355  df-cmp 23644  df-fil 24104  df-ufil 24159  df-ufl 24160  df-flim 24197  df-fcls 24199

This theorem is used by:  tmachlem-exagreecover  47839