Theorem tmachlem-agreesn 48160

Description: Scans for all tapes of a single agreement set are identical. (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-agreesn	⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆 “ (𝐴‘𝑎)) = {(𝑆‘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-agreesn

Step	Hyp	Ref	Expression
1		nfv 1949	. . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑎 ∈ 𝑇)
2		tmach.scanmap	. . . . 5 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
3	2	ffund 6733	. . . 4 ⊢ (𝜑 → Fun 𝑆)
4	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → Fun 𝑆)
5		tmach.agreement	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))
6		fveq2 6905	. . . . . . . . 9 ⊢ (𝑧 = 𝑎 → (𝐴‘𝑧) = (𝐴‘𝑎))
7		fveq2 6905	. . . . . . . . . 10 ⊢ (𝑧 = 𝑎 → (𝑆‘𝑧) = (𝑆‘𝑎))
8	7	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑧 = 𝑎 → ((𝑆‘𝑦) = (𝑆‘𝑧) ↔ (𝑆‘𝑦) = (𝑆‘𝑎)))
9	6, 8	raleqbidv 3338	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → (∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧) ↔ ∀𝑦 ∈ (𝐴‘𝑎)(𝑆‘𝑦) = (𝑆‘𝑎)))
10	9	cbvralvw 3244	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧) ↔ ∀𝑎 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑎)(𝑆‘𝑦) = (𝑆‘𝑎))
11	5, 10	sylib 221	. . . . . 6 ⊢ (𝜑 → ∀𝑎 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑎)(𝑆‘𝑦) = (𝑆‘𝑎))
12	11	r19.21bi 3258	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∀𝑦 ∈ (𝐴‘𝑎)(𝑆‘𝑦) = (𝑆‘𝑎))
13	12	r19.21bi 3258	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ (𝐴‘𝑎)) → (𝑆‘𝑦) = (𝑆‘𝑎))
14		fvex 6918	. . . . 5 ⊢ (𝑆‘𝑎) ∈ V
15	14	elsn2 4626	. . . 4 ⊢ ((𝑆‘𝑦) ∈ {(𝑆‘𝑎)} ↔ (𝑆‘𝑦) = (𝑆‘𝑎))
16	13, 15	sylibr 237	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ (𝐴‘𝑎)) → (𝑆‘𝑦) ∈ {(𝑆‘𝑎)})
17	1, 4, 16	funimassd 6971	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆 “ (𝐴‘𝑎)) ⊆ {(𝑆‘𝑎)})
18		tmach.finalph	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin)
19		tmach.exindex	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
20		tmach.tapelist	. . . . 5 ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))
21		tmach.agreemap	. . . . 5 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))
22	18, 19, 20, 2, 21, 5	tmachlem-agreeself 48149	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝐴‘𝑎))
23	2	fdmd 6739	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝑇)
24	23	eleq2d 2850	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ dom 𝑆 ↔ 𝑎 ∈ 𝑇))
25	24	biimpar 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ dom 𝑆)
26		funfvima 7256	. . . . 5 ⊢ ((Fun 𝑆 ∧ 𝑎 ∈ dom 𝑆) → (𝑎 ∈ (𝐴‘𝑎) → (𝑆‘𝑎) ∈ (𝑆 “ (𝐴‘𝑎))))
27	4, 25, 26	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑎 ∈ (𝐴‘𝑎) → (𝑆‘𝑎) ∈ (𝑆 “ (𝐴‘𝑎))))
28	22, 27	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆‘𝑎) ∈ (𝑆 “ (𝐴‘𝑎)))
29	28	snssd 4747	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → {(𝑆‘𝑎)} ⊆ (𝑆 “ (𝐴‘𝑎)))
30	17, 29	eqssd 3930	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆 “ (𝐴‘𝑎)) = {(𝑆‘𝑎)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ∩ cin 4092  𝒫 cpw 4560  {csn 4588   ↦ cmpt 5189  dom cdm 5655   ↾ cres 5657   “ cima 5658  Fun wfun 6549  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872  Fincfn 8998

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-ov 7443

This theorem is used by:  tmachlem-agreefin  48161