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Theorem lmbr2 21003
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 21002 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢))))
3 uzf 11650 . . . . . . . 8 :ℤ⟶𝒫 ℤ
4 ffn 6012 . . . . . . . 8 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
5 reseq2 5361 . . . . . . . . . 10 (𝑧 = (ℤ𝑗) → (𝐹𝑧) = (𝐹 ↾ (ℤ𝑗)))
6 id 22 . . . . . . . . . 10 (𝑧 = (ℤ𝑗) → 𝑧 = (ℤ𝑗))
75, 6feq12d 6000 . . . . . . . . 9 (𝑧 = (ℤ𝑗) → ((𝐹𝑧):𝑧𝑢 ↔ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢))
87rexrn 6327 . . . . . . . 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 7837 . . . . . . . . . . 11 (𝐹 ∈ (𝑋pm ℂ) → Fun 𝐹)
1110ad2antrl 763 . . . . . . . . . 10 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → Fun 𝐹)
12 ffvresb 6360 . . . . . . . . . 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 3047 . . . . . . . 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 481 . . . . . . . . 9 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → 𝑀 ∈ ℤ)
17 lmbr2.4 . . . . . . . . . 10 𝑍 = (ℤ𝑀)
1817rexuz3 14038 . . . . . . . . 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 271 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
219, 20syl5bb 272 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
2221imbi2d 330 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → ((𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢) ↔ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
2322ralbidv 2982 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
2423pm5.32da 672 . . 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 1038 . . 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 1038 . . 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 303 . 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 268 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wrex 2909  𝒫 cpw 4136   class class class wbr 4623  dom cdm 5084  ran crn 5085  cres 5086  Fun wfun 5851   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  pm cpm 7818  cc 9894  cz 11337  cuz 11647  TopOnctopon 20655  𝑡clm 20970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-pre-lttri 9970  ax-pre-lttrn 9971
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-po 5005  df-so 5006  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-er 7702  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-neg 10229  df-z 11338  df-uz 11648  df-top 20639  df-topon 20656  df-lm 20973
This theorem is referenced by:  lmbrf  21004  lmcvg  21006  lmres  21044  lmcls  21046  lmcnp  21048
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