Theorem tmachlem-franscan 47875

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 6699	. . 3 ⊢ (𝜑 → 𝑆 Fn 𝑇)
3		fnima 6658	. . 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 47872	. . 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 7239	. . . . 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 3932	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎)) ∧ 𝑖 ∈ 𝑎) → 𝑖 ∈ ran 𝐴)
20	5, 6, 7, 1, 8, 9	tmachlem-agreefin 47874	. . . . . . 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 9310	. . . . 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 1968	. 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 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ∪ ciun 4951   ↦ cmpt 5186  ran crn 5649   ↾ cres 5650   “ cima 5651   Fn wfn 6523  ⟶wf 6524  ‘cfv 6528  (class class class)co 7409   ↑_m cmap 8826  Fincfn 8952

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 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735  ax-ac2 10498

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 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-isom 6537  df-riota 7366  df-ov 7412  df-oprab 7413  df-mpo 7414  df-rpss 7723  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8455  df-2o 8456  df-oadd 8459  df-omul 8460  df-er 8696  df-map 8828  df-ixp 8905  df-en 8953  df-dom 8954  df-fin 8956  df-fi 9381  df-wdom 9537  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10152  df-topgen 17561  df-pt 17562  df-fbas 21622  df-fg 21623  df-top 23159  df-topon 23176  df-bases 23211  df-cld 23284  df-ntr 23285  df-cls 23286  df-nei 23363  df-cmp 23652  df-fil 24112  df-ufil 24167  df-ufl 24168  df-flim 24205  df-fcls 24207

This theorem is used by:  tmachfullfin  47877