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Theorem lmbr2 23283
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 23282 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢))))
3 uzf 12879 . . . . . . . 8 :ℤ⟶𝒫 ℤ
4 ffn 6737 . . . . . . . 8 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
5 reseq2 5995 . . . . . . . . . 10 (𝑧 = (ℤ𝑗) → (𝐹𝑧) = (𝐹 ↾ (ℤ𝑗)))
6 id 22 . . . . . . . . . 10 (𝑧 = (ℤ𝑗) → 𝑧 = (ℤ𝑗))
75, 6feq12d 6725 . . . . . . . . 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 8886 . . . . . . . . . . 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 3177 . . . . . . . 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 15384 . . . . . . . . 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 3176 . . . 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 1088 . . 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 1088 . . 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 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  𝒫 cpw 4605   class class class wbr 5148  dom cdm 5689  ran crn 5690  cres 5691  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  pm cpm 8866  cc 11151  cz 12611  cuz 12876  TopOnctopon 22932  𝑡clm 23250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-neg 11493  df-z 12612  df-uz 12877  df-top 22916  df-topon 22933  df-lm 23253
This theorem is referenced by:  lmbrf  23284  lmcvg  23286  lmres  23324  lmcls  23326  lmcnp  23328  lmbr3v  45701
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