Theorem tmachlem-extpcover 47758

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 47751	. 2 ⊢ (𝜑 → (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) ∈ Comp)
8	1, 2, 3, 4, 5, 6	tmachlem-extapes 47747	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ V)
9		ssrab2 4031	. . . . . . . 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 5293	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
12	11	ralrimivw 3160	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
13		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑧𝑇
14	13	mptfnf 6671	. . . . 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 6629	. . . 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 47755	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑇) → (𝐴‘𝑏) ∈ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)))
19	18	ralrimiva 3156	. . 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 47752	. . 3 ⊢ (𝜑 → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = 𝑇)
23	1, 2, 3, 4, 5, 6	tmachlem-exlargecover 47757	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 = 𝑇)
24	22, 23	eqtr4d 2800	. 2 ⊢ (𝜑 → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ ran 𝐴)
25		eqid 2762	. . 3 ⊢ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈))
26	25	cmpcov 23613	. 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 3078  ∃wrex 3088  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870   ↦ cmpt 5190  ran crn 5660   ↾ cres 5661   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ↑_m cmap 8829  Fincfn 8955  ∏_tcpt 17525  Compccmp 23610

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-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-ac2 10468

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-rpss 7727  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-omul 8463  df-er 8699  df-map 8831  df-ixp 8908  df-en 8956  df-dom 8957  df-fin 8959  df-fi 9384  df-wdom 9540  df-dju 9909  df-card 9947  df-acn 9950  df-ac 10122  df-topgen 17530  df-pt 17531  df-fbas 21581  df-fg 21582  df-top 23118  df-topon 23135  df-bases 23170  df-cld 23243  df-ntr 23244  df-cls 23245  df-nei 23322  df-cmp 23611  df-fil 24071  df-ufil 24126  df-ufl 24127  df-flim 24164  df-fcls 24166

This theorem is used by:  tmachlem-exagreecover  47759