Theorem tmachlem-agreesn 47957

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 1946	. . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑎 ∈ 𝑇)
2		tmach.scanmap	. . . . 5 ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))
3	2	ffund 6714	. . . 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 6885	. . . . . . . . 9 ⊢ (𝑧 = 𝑎 → (𝐴‘𝑧) = (𝐴‘𝑎))
7		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑧 = 𝑎 → (𝑆‘𝑧) = (𝑆‘𝑎))
8	7	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑧 = 𝑎 → ((𝑆‘𝑦) = (𝑆‘𝑧) ↔ (𝑆‘𝑦) = (𝑆‘𝑎)))
9	6, 8	raleqbidv 3334	. . . . . . . 8 ⊢ (𝑧 = 𝑎 → (∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧) ↔ ∀𝑦 ∈ (𝐴‘𝑎)(𝑆‘𝑦) = (𝑆‘𝑎)))
10	9	cbvralvw 3240	. . . . . . 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 3254	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → ∀𝑦 ∈ (𝐴‘𝑎)(𝑆‘𝑦) = (𝑆‘𝑎))
13	12	r19.21bi 3254	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑇) ∧ 𝑦 ∈ (𝐴‘𝑎)) → (𝑆‘𝑦) = (𝑆‘𝑎))
14		fvex 6898	. . . . 5 ⊢ (𝑆‘𝑎) ∈ V
15	14	elsn2 4625	. . . 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 6951	. 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 47946	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝐴‘𝑎))
23	2	fdmd 6720	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝑇)
24	23	eleq2d 2846	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ dom 𝑆 ↔ 𝑎 ∈ 𝑇))
25	24	biimpar 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ dom 𝑆)
26		funfvima 7236	. . . . 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 4746	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → {(𝑆‘𝑎)} ⊆ (𝑆 “ (𝐴‘𝑎)))
30	17, 29	eqssd 3925	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆 “ (𝐴‘𝑎)) = {(𝑆‘𝑎)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ∩ cin 4089  𝒫 cpw 4556  {csn 4583   ↦ cmpt 5185  dom cdm 5650   ↾ 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-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-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423

This theorem is used by:  tmachlem-agreefin  47958