Theorem tmachlem-agreefin 47761

Description: Any agreement set has a finite (singleton) list of possible scans. (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-agreefin	⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐴) → (𝑆 “ 𝑏) ∈ Fin)

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

Proof of Theorem tmachlem-agreefin

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6575	. . . . 5 ⊢ Fun (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))})
2		tmach.agreemap	. . . . . 6 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
3	2	funeqd 6559	. . . . 5 ⊢ (𝜑 → (Fun 𝐴 ↔ Fun (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))})))
4	1, 3	mpbiri 261	. . . 4 ⊢ (𝜑 → Fun 𝐴)
5		elrnrexdm 7085	. . . 4 ⊢ (Fun 𝐴 → (𝑏 ∈ ran 𝐴 → ∃𝑎 ∈ dom 𝐴 𝑏 = (𝐴‘𝑎)))
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → (𝑏 ∈ ran 𝐴 → ∃𝑎 ∈ dom 𝐴 𝑏 = (𝐴‘𝑎)))
7	6	imp 412	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐴) → ∃𝑎 ∈ dom 𝐴 𝑏 = (𝐴‘𝑎))
8		simprr 785	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → 𝑏 = (𝐴‘𝑎))
9	8	imaeq2d 6060	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → (𝑆 “ 𝑏) = (𝑆 “ (𝐴‘𝑎)))
10		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → 𝜑)
11		simprl 783	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → 𝑎 ∈ dom 𝐴)
12	2	dmeqd 5893	. . . . . . . 8 ⊢ (𝜑 → dom 𝐴 = dom (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
13		tmach.finalph	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ Fin)
14		tmach.exindex	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ V)
15		tmach.tapelist	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
16		tmach.scanmap	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
17		tmach.agreement	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))
18	13, 14, 15, 16, 2, 17	tmachlem-extapes 47747	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ V)
19		ssrab2 4031	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ⊆ 𝑇
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ⊆ 𝑇)
21	18, 20	ssexd 5293	. . . . . . . . . 10 ⊢ (𝜑 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
22	21	ralrimivw 3160	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
23		dmmptg 6242	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V → dom (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}) = 𝑇)
24	22, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → dom (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}) = 𝑇)
25	12, 24	eqtrd 2797	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = 𝑇)
26	25	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → dom 𝐴 = 𝑇)
27	11, 26	eleqtrd 2864	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → 𝑎 ∈ 𝑇)
28	13, 14, 15, 16, 2, 17	tmachlem-agreesn 47760	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆 “ (𝐴‘𝑎)) = {(𝑆‘𝑎)})
29	10, 27, 28	syl2anc 596	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → (𝑆 “ (𝐴‘𝑎)) = {(𝑆‘𝑎)})
30	9, 29	eqtrd 2797	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → (𝑆 “ 𝑏) = {(𝑆‘𝑎)})
31		snfi 9053	. . 3 ⊢ {(𝑆‘𝑎)} ∈ Fin
32	30, 31	eqeltrdi 2870	. 2 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → (𝑆 “ 𝑏) ∈ Fin)
33	7, 32	rexlimddv 3171	1 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐴) → (𝑆 “ 𝑏) ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560  {csn 4587   ↦ cmpt 5190  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662  Fun wfun 6531  ⟶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-sep 5255  ax-nul 5267  ax-pr 5402

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-ral 3079  df-rex 3089  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-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-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-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-ov 7419  df-om 7866  df-1o 8458  df-en 8956  df-fin 8959

This theorem is used by:  tmachlem-franscan  47762