Theorem tmachlem-exagreecover 47956

Description: Particular properties of the finite cover of agreesets. (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-exagreecover	⊢ (𝜑 → ∃𝑎(𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎))

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

Proof of Theorem tmachlem-exagreecover

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tmach.finalph	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Fin)
2		tmach.exindex	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
3		tmach.tapelist	. . . 4 ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
4		tmach.scanmap	. . . 4 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
5		tmach.agreemap	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
6		tmach.agreement	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))
7	1, 2, 3, 4, 5, 6	tmachlem-extpcover 47955	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin)∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)
8		df-rex 3087	. . 3 ⊢ (∃𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin)∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎 ↔ ∃𝑎(𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎))
9	7, 8	sylib 221	. 2 ⊢ (𝜑 → ∃𝑎(𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎))
10		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)) → 𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin))
11	10	elin1d 4101	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)) → 𝑎 ∈ 𝒫 ran 𝐴)
12	11	elpwid 4565	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)) → 𝑎 ⊆ ran 𝐴)
13	10	elin2d 4102	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)) → 𝑎 ∈ Fin)
14	1, 2, 3, 4, 5, 6	tmachlem-tpbase 47949	. . . . . . 7 ⊢ (𝜑 → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = 𝑇)
15	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)) → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = 𝑇)
16		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)) → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)
17	15, 16	eqtr3d 2797	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)) → 𝑇 = ∪ 𝑎)
18	12, 13, 17	3jca 1146	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)) → (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎))
19	18	ex 418	. . 3 ⊢ (𝜑 → ((𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎) → (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)))
20	19	eximdv 1949	. 2 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin) ∧ ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎) → ∃𝑎(𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)))
21	9, 20	mpd 16	1 ⊢ (𝜑 → ∃𝑎(𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  ∪ cuni 4866   ↦ cmpt 5185  ran crn 5651   ↾ cres 5652  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  Fincfn 8973  ∏_tcpt 17609

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-ac2 10541

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rpss 7739  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-omul 8481  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-fin 8977  df-fi 9403  df-wdom 9559  df-dju 9982  df-card 10020  df-acn 10023  df-ac 10195  df-topgen 17614  df-pt 17615  df-fbas 21675  df-fg 21676  df-top 23212  df-topon 23229  df-bases 23264  df-cld 23337  df-ntr 23338  df-cls 23339  df-nei 23416  df-cmp 23705  df-fil 24165  df-ufil 24220  df-ufl 24221  df-flim 24258  df-fcls 24260

This theorem is used by:  tmachlem-franscan  47959