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Theorem lmbr2 21435
 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 21434 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢))))
3 uzf 11972 . . . . . . . 8 :ℤ⟶𝒫 ℤ
4 ffn 6279 . . . . . . . 8 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
5 reseq2 5625 . . . . . . . . . 10 (𝑧 = (ℤ𝑗) → (𝐹𝑧) = (𝐹 ↾ (ℤ𝑗)))
6 id 22 . . . . . . . . . 10 (𝑧 = (ℤ𝑗) → 𝑧 = (ℤ𝑗))
75, 6feq12d 6267 . . . . . . . . 9 (𝑧 = (ℤ𝑗) → ((𝐹𝑧):𝑧𝑢 ↔ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢))
87rexrn 6611 . . . . . . . 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 8143 . . . . . . . . . . 11 (𝐹 ∈ (𝑋pm ℂ) → Fun 𝐹)
1110ad2antrl 721 . . . . . . . . . 10 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → Fun 𝐹)
12 ffvresb 6644 . . . . . . . . . 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 3263 . . . . . . . 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 474 . . . . . . . . 9 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → 𝑀 ∈ ℤ)
17 lmbr2.4 . . . . . . . . . 10 𝑍 = (ℤ𝑀)
1817rexuz3 14466 . . . . . . . . 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 274 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ𝑗)):(ℤ𝑗)⟶𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
219, 20syl5bb 275 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
2221imbi2d 332 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → ((𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢) ↔ (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
2322ralbidv 3196 . . . 4 ((𝜑 ∧ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋)) → (∀𝑢𝐽 (𝑃𝑢 → ∃𝑧 ∈ ran ℤ(𝐹𝑧):𝑧𝑢) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
2423pm5.32da 576 . . 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 1115 . . 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 1115 . . 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 306 . 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 271 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 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3118  ∃wrex 3119  𝒫 cpw 4379   class class class wbr 4874  dom cdm 5343  ran crn 5344   ↾ cres 5345  Fun wfun 6118   Fn wfn 6119  ⟶wf 6120  ‘cfv 6124  (class class class)co 6906   ↑pm cpm 8124  ℂcc 10251  ℤcz 11705  ℤ≥cuz 11969  TopOnctopon 21086  ⇝𝑡clm 21402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-pre-lttri 10327  ax-pre-lttrn 10328 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-po 5264  df-so 5265  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-er 8010  df-pm 8126  df-en 8224  df-dom 8225  df-sdom 8226  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-neg 10589  df-z 11706  df-uz 11970  df-top 21070  df-topon 21087  df-lm 21405 This theorem is referenced by:  lmbrf  21436  lmcvg  21438  lmres  21476  lmcls  21478  lmcnp  21480  lmbr3v  40773
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