Theorem tmachlem-franscan 47762

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 6707	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝑇)
3		fnima 6666	. . 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 47759	. . 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 6060	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → (𝑆 “ 𝑇) = (𝑆 “ ∪ 𝑎))
13		imauni 7246	. . . . 5 ⊢ (𝑆 “ ∪ 𝑎) = ∪ 𝑖 ∈ 𝑎 (𝑆 “ 𝑖)
14	12, 13	eqtrdi 2813	. . . 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 3935	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) ∧ 𝑖 ∈ 𝑎) → 𝑖 ∈ ran 𝐴)
20	5, 6, 7, 1, 8, 9	tmachlem-agreefin 47761	. . . . . . 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 3156	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → ∀𝑖 ∈ 𝑎 (𝑆 “ 𝑖) ∈ Fin)
23		iunfi 9313	. . . . 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 2862	. . 3 ⊢ ((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) → (𝑆 “ 𝑇) ∈ Fin)
26	10, 25	exlimddv 1968	. 2 ⊢ (𝜑 → (𝑆 “ 𝑇) ∈ Fin)
27	4, 26	eqeltrrd 2863	1 ⊢ (𝜑 → ran 𝑆 ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5190  ran crn 5660   ↾ cres 5661   “ cima 5662   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ↑_m cmap 8829  Fincfn 8955

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:  tmachfullfin  47764