Theorem tmachlem-agreefin 47958

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 6578	. . . . 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 6561	. . . . 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 7089	. . . 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 6051	. . . 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 5886	. . . . . . . 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 47944	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ V)
19		ssrab2 4005	. . . . . . . . . . . 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 5285	. . . . . . . . . 10 ⊢ (𝜑 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
22	21	ralrimivw 3158	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} ∈ V)
23		dmmptg 6243	. . . . . . . . 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 2795	. . . . . . 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 2862	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → 𝑎 ∈ 𝑇)
28	13, 14, 15, 16, 2, 17	tmachlem-agreesn 47957	. . . . 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 2795	. . 3 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → (𝑆 “ 𝑏) = {(𝑆‘𝑎)})
31		snfi 9071	. . 3 ⊢ {(𝑆‘𝑎)} ∈ Fin
32	30, 31	eqeltrdi 2868	. 2 ⊢ (((𝜑 ∧ 𝑏 ∈ ran 𝐴) ∧ (𝑎 ∈ dom 𝐴 ∧ 𝑏 = (𝐴‘𝑎))) → (𝑆 “ 𝑏) ∈ Fin)
33	7, 32	rexlimddv 3169	1 ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐴) → (𝑆 “ 𝑏) ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  {csn 4583   ↦ cmpt 5185  dom cdm 5650  ran crn 5651   ↾ cres 5652   “ cima 5653  Fun wfun 6532  ⟶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-sep 5248  ax-nul 5259  ax-pr 5390

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-ral 3077  df-rex 3087  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-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-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-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-ov 7423  df-om 7878  df-1o 8476  df-en 8974  df-fin 8977

This theorem is used by:  tmachlem-franscan  47959