Theorem tmachlem-agreeself 47946

Description: Any tape belongs to its own agreement set. (Contributed by Ender Ting, 27-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-agreeself	⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝐴‘𝑎))

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

Proof of Theorem tmachlem-agreeself

Step	Hyp	Ref	Expression
1		reseq1 5963	. . . 4 ⊢ (𝑦 = 𝑎 → (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎)))
2		tbtru 1577	. . . 4 ⊢ ((𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎)) ↔ ((𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎)) ↔ ⊤))
3	1, 2	sylib 221	. . 3 ⊢ (𝑦 = 𝑎 → ((𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎)) ↔ ⊤))
4		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ 𝑇)
5		trud 1579	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ⊤)
6	3, 4, 5	elrabd 3646	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))})
7		fveq2 6885	. . . . . 6 ⊢ (𝑧 = 𝑎 → (𝑆‘𝑧) = (𝑆‘𝑎))
8	7	reseq2d 5969	. . . . 5 ⊢ (𝑧 = 𝑎 → (𝑦 ↾ (𝑆‘𝑧)) = (𝑦 ↾ (𝑆‘𝑎)))
9		id 23	. . . . . 6 ⊢ (𝑧 = 𝑎 → 𝑧 = 𝑎)
10	9, 7	reseq12d 5970	. . . . 5 ⊢ (𝑧 = 𝑎 → (𝑧 ↾ (𝑆‘𝑧)) = (𝑎 ↾ (𝑆‘𝑎)))
11	8, 10	eqeq12d 2776	. . . 4 ⊢ (𝑧 = 𝑎 → ((𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧)) ↔ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))))
12	11	rabbidv 3419	. . 3 ⊢ (𝑧 = 𝑎 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))} = {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))})
13		tmach.agreemap	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
14	13	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
15		tmach.finalph	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Fin)
16		tmach.exindex	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V)
17		tmach.tapelist	. . . . . 6 ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
18		tmach.scanmap	. . . . . 6 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
19		tmach.agreement	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))
20	15, 16, 17, 18, 13, 19	tmachlem-extapes 47944	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ V)
21		ssrab2 4005	. . . . . 6 ⊢ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))} ⊆ 𝑇
22	21	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))} ⊆ 𝑇)
23	20, 22	ssexd 5285	. . . 4 ⊢ (𝜑 → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))} ∈ V)
24	23	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))} ∈ V)
25	12, 14, 4, 24	fvmptd4 7018	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝐴‘𝑎) = {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑎)) = (𝑎 ↾ (𝑆‘𝑎))})
26	6, 25	eleqtrrd 2863	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝐴‘𝑎))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556   ↦ cmpt 5185   ↾ cres 5652  ⟶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-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-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423

This theorem is used by:  tmachlem-exlargecover  47954  tmachlem-agreesn  47957