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Theorem lmbr2 23267
Description: Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Mario Carneiro, 14-Nov-2013.)
Hypotheses
Ref Expression
lmbr.2 (𝜑𝐽 ∈ (TopOn‘𝑋))
lmbr2.4 𝑍 = (ℤ𝑀)
lmbr2.5 (𝜑𝑀 ∈ ℤ)
Assertion
Ref Expression
lmbr2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
Distinct variable groups:   𝑗,𝑘,𝑢,𝐹   𝑗,𝐽,𝑘,𝑢   𝜑,𝑗,𝑘,𝑢   𝑗,𝑍,𝑘,𝑢   𝑗,𝑀   𝑃,𝑗,𝑘,𝑢   𝑗,𝑋,𝑘,𝑢
Allowed substitution hints:   𝑀(𝑢,𝑘)

Proof of Theorem lmbr2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 lmbr.2 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
21lmbr 23266 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢))))
3 uzf 12881 . . . . . . . 8 :ℤ⟶𝒫 ℤ
4 ffn 6736 . . . . . . . 8 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
5 reseq2 5992 . . . . . . . . . 10 (𝑧 = (ℤ𝑗) → (𝐹𝑧) = (𝐹 ↾ (ℤ𝑗)))
6 id 22 . . . . . . . . . 10 (𝑧 = (ℤ𝑗) → 𝑧 = (ℤ𝑗))
75, 6feq12d 6724 . . . . . . . . 9 (𝑧 = (ℤ𝑗) → ((𝐹𝑧):𝑧𝑢 ↔ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢))
87rexrn 7107 . . . . . . . 8 (ℤ Fn ℤ → (∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢))
93, 4, 8mp2b 10 . . . . . . 7 (∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢)
10 pmfun 8887 . . . . . . . . . . 11 (𝐹 ∈ (𝑋pm ℂ) → Fun 𝐹)
1110ad2antrl 728 . . . . . . . . . 10 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → Fun 𝐹)
12 ffvresb 7145 . . . . . . . . . 10 (Fun 𝐹 → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢 ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
1311, 12syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → ((𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢 ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
1413rexbidv 3179 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
15 lmbr2.5 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
1615adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → 𝑀 ∈ ℤ)
17 lmbr2.4 . . . . . . . . . 10 𝑍 = (ℤ𝑀)
1817rexuz3 15387 . . . . . . . . 9 (𝑀 ∈ ℤ → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
1916, 18syl 17 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
2014, 19bitr4d 282 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
219, 20bitrid 283 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
2221imbi2d 340 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → ((𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢) ↔ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
2322ralbidv 3178 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
2423pm5.32da 579 . . 3 (𝜑 → (((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢)) ↔ ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
25 df-3an 1089 . . 3 ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢)) ↔ ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢)))
26 df-3an 1089 . . 3 ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))) ↔ ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋) ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
2724, 25, 263bitr4g 314 . 2 (𝜑 → ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢)) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
282, 27bitrd 279 1 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  𝒫 cpw 4600   class class class wbr 5143  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  pm cpm 8867  cc 11153  cz 12613  cuz 12878  TopOnctopon 22916  𝑡clm 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-neg 11495  df-z 12614  df-uz 12879  df-top 22900  df-topon 22917  df-lm 23237
This theorem is referenced by:  lmbrf  23268  lmcvg  23270  lmres  23308  lmcls  23310  lmcnp  23312  lmbr3v  45760
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