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Theorem lmbr3 40617
Description: Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypotheses
Ref Expression
lmbr3.1 𝑘𝐹
lmbr3.2 (𝜑𝐽 ∈ (TopOn‘𝑋))
Assertion
Ref Expression
lmbr3 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
Distinct variable groups:   𝑗,𝐹,𝑢   𝑢,𝐽   𝑢,𝑃   𝑗,𝑘,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘)   𝑃(𝑗,𝑘)   𝐹(𝑘)   𝐽(𝑗,𝑘)   𝑋(𝑢,𝑗,𝑘)

Proof of Theorem lmbr3
Dummy variables 𝑖 𝑙 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmbr3.2 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
21lmbr3v 40615 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑣𝐽 (𝑃𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣)))))
3 eleq2w 2828 . . . . 5 (𝑣 = 𝑢 → (𝑃𝑣𝑃𝑢))
4 eleq2w 2828 . . . . . . . 8 (𝑣 = 𝑢 → ((𝐹𝑙) ∈ 𝑣 ↔ (𝐹𝑙) ∈ 𝑢))
54anbi2d 622 . . . . . . 7 (𝑣 = 𝑢 → ((𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣) ↔ (𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢)))
65rexralbidv 3205 . . . . . 6 (𝑣 = 𝑢 → (∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣) ↔ ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢)))
7 fveq2 6375 . . . . . . . . 9 (𝑖 = 𝑗 → (ℤ𝑖) = (ℤ𝑗))
87raleqdv 3292 . . . . . . . 8 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ ∀𝑙 ∈ (ℤ𝑗)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢)))
9 nfcv 2907 . . . . . . . . . . 11 𝑘𝑙
10 lmbr3.1 . . . . . . . . . . . 12 𝑘𝐹
1110nfdm 5536 . . . . . . . . . . 11 𝑘dom 𝐹
129, 11nfel 2920 . . . . . . . . . 10 𝑘 𝑙 ∈ dom 𝐹
1310, 9nffv 6385 . . . . . . . . . . 11 𝑘(𝐹𝑙)
14 nfcv 2907 . . . . . . . . . . 11 𝑘𝑢
1513, 14nfel 2920 . . . . . . . . . 10 𝑘(𝐹𝑙) ∈ 𝑢
1612, 15nfan 1998 . . . . . . . . 9 𝑘(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢)
17 nfv 2009 . . . . . . . . 9 𝑙(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)
18 eleq1w 2827 . . . . . . . . . 10 (𝑙 = 𝑘 → (𝑙 ∈ dom 𝐹𝑘 ∈ dom 𝐹))
19 fveq2 6375 . . . . . . . . . . 11 (𝑙 = 𝑘 → (𝐹𝑙) = (𝐹𝑘))
2019eleq1d 2829 . . . . . . . . . 10 (𝑙 = 𝑘 → ((𝐹𝑙) ∈ 𝑢 ↔ (𝐹𝑘) ∈ 𝑢))
2118, 20anbi12d 624 . . . . . . . . 9 (𝑙 = 𝑘 → ((𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
2216, 17, 21cbvral 3315 . . . . . . . 8 (∀𝑙 ∈ (ℤ𝑗)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))
238, 22syl6bb 278 . . . . . . 7 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
2423cbvrexv 3320 . . . . . 6 (∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))
256, 24syl6bb 278 . . . . 5 (𝑣 = 𝑢 → (∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
263, 25imbi12d 335 . . . 4 (𝑣 = 𝑢 → ((𝑃𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣)) ↔ (𝑃𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
2726cbvralv 3319 . . 3 (∀𝑣𝐽 (𝑃𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣)) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
28273anbi3i 1198 . 2 ((𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑣𝐽 (𝑃𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣))) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
292, 28syl6bb 278 1 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wcel 2155  wnfc 2894  wral 3055  wrex 3056   class class class wbr 4809  dom cdm 5277  cfv 6068  (class class class)co 6842  pm cpm 8061  cc 10187  cz 11624  cuz 11886  TopOnctopon 20994  𝑡clm 21310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-addrcl 10250  ax-rnegex 10260  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-er 7947  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-neg 10523  df-z 11625  df-uz 11887  df-top 20978  df-topon 20995  df-lm 21313
This theorem is referenced by:  xlimbr  40691  xlimmnfvlem1  40696  xlimmnfvlem2  40697  xlimpnfvlem1  40700  xlimpnfvlem2  40701
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