Theorem tmachlem-franscan 47959

Description: There is a finite number of different scan sets. (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-franscan	⊢ (𝜑 → ran 𝑆 ∈ Fin)

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-franscan

Dummy variables 𝑎 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tmach.scanmap	. . . 4 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
2	1	ffnd 6710	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝑇)
3		fnima 6669	. . 3 ⊢ (𝑆 Fn 𝑇 → (𝑆 “ 𝑇) = ran 𝑆)
4	2, 3	syl 18	. 2 ⊢ (𝜑 → (𝑆 “ 𝑇) = ran 𝑆)
5		tmach.finalph	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Fin)
6		tmach.exindex	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
7		tmach.tapelist	. . . 4 ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
8		tmach.agreemap	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
9		tmach.agreement	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))
10	5, 6, 7, 1, 8, 9	tmachlem-exagreecover 47956	. . 3 ⊢ (𝜑 → ∃𝑎(𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎))
11		simpr3 1215	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → 𝑇 = ∪ 𝑎)
12	11	imaeq2d 6051	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → (𝑆 “ 𝑇) = (𝑆 “ ∪ 𝑎))
13		imauni 7250	. . . . 5 ⊢ (𝑆 “ ∪ 𝑎) = ∪ 𝑖 ∈ 𝑎 (𝑆 “ 𝑖)
14	12, 13	eqtrdi 2811	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → (𝑆 “ 𝑇) = ∪ 𝑖 ∈ 𝑎 (𝑆 “ 𝑖))
15		simpr2 1214	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → 𝑎 ∈ Fin)
16		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) ∧ 𝑖 ∈ 𝑎) → 𝜑)
17		simplr1 1234	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) ∧ 𝑖 ∈ 𝑎) → 𝑎 ⊆ ran 𝐴)
18		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) ∧ 𝑖 ∈ 𝑎) → 𝑖 ∈ 𝑎)
19	17, 18	sseldd 3909	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) ∧ 𝑖 ∈ 𝑎) → 𝑖 ∈ ran 𝐴)
20	5, 6, 7, 1, 8, 9	tmachlem-agreefin 47958	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ran 𝐴) → (𝑆 “ 𝑖) ∈ Fin)
21	16, 19, 20	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) ∧ 𝑖 ∈ 𝑎) → (𝑆 “ 𝑖) ∈ Fin)
22	21	ralrimiva 3154	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → ∀𝑖 ∈ 𝑎 (𝑆 “ 𝑖) ∈ Fin)
23		iunfi 9332	. . . . 5 ⊢ ((𝑎 ∈ Fin ∧ ∀𝑖 ∈ 𝑎 (𝑆 “ 𝑖) ∈ Fin) → ∪ 𝑖 ∈ 𝑎 (𝑆 “ 𝑖) ∈ Fin)
24	15, 22, 23	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → ∪ 𝑖 ∈ 𝑎 (𝑆 “ 𝑖) ∈ Fin)
25	14, 24	eqeltrd 2860	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → (𝑆 “ 𝑇) ∈ Fin)
26	10, 25	exlimddv 1967	. 2 ⊢ (𝜑 → (𝑆 “ 𝑇) ∈ Fin)
27	4, 26	eqeltrrd 2861	1 ⊢ (𝜑 → ran 𝑆 ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5185  ran crn 5651   ↾ cres 5652   “ cima 5653   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  Fincfn 8973

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:  tmachfullfin  47961