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Theorem lmbr3 42180
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 42178 . 2 (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑣𝐽 (𝑃𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣)))))
3 eleq2w 2894 . . . . 5 (𝑣 = 𝑢 → (𝑃𝑣𝑃𝑢))
4 eleq2w 2894 . . . . . . . 8 (𝑣 = 𝑢 → ((𝐹𝑙) ∈ 𝑣 ↔ (𝐹𝑙) ∈ 𝑢))
54anbi2d 630 . . . . . . 7 (𝑣 = 𝑢 → ((𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣) ↔ (𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢)))
65rexralbidv 3288 . . . . . 6 (𝑣 = 𝑢 → (∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣) ↔ ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢)))
7 fveq2 6646 . . . . . . . . 9 (𝑖 = 𝑗 → (ℤ𝑖) = (ℤ𝑗))
87raleqdv 3398 . . . . . . . 8 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ ∀𝑙 ∈ (ℤ𝑗)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢)))
9 nfcv 2973 . . . . . . . . . . 11 𝑘𝑙
10 lmbr3.1 . . . . . . . . . . . 12 𝑘𝐹
1110nfdm 5799 . . . . . . . . . . 11 𝑘dom 𝐹
129, 11nfel 2987 . . . . . . . . . 10 𝑘 𝑙 ∈ dom 𝐹
1310, 9nffv 6656 . . . . . . . . . . 11 𝑘(𝐹𝑙)
14 nfcv 2973 . . . . . . . . . . 11 𝑘𝑢
1513, 14nfel 2987 . . . . . . . . . 10 𝑘(𝐹𝑙) ∈ 𝑢
1612, 15nfan 1900 . . . . . . . . 9 𝑘(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢)
17 nfv 1915 . . . . . . . . 9 𝑙(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)
18 eleq1w 2893 . . . . . . . . . 10 (𝑙 = 𝑘 → (𝑙 ∈ dom 𝐹𝑘 ∈ dom 𝐹))
19 fveq2 6646 . . . . . . . . . . 11 (𝑙 = 𝑘 → (𝐹𝑙) = (𝐹𝑘))
2019eleq1d 2895 . . . . . . . . . 10 (𝑙 = 𝑘 → ((𝐹𝑙) ∈ 𝑢 ↔ (𝐹𝑘) ∈ 𝑢))
2118, 20anbi12d 632 . . . . . . . . 9 (𝑙 = 𝑘 → ((𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
2216, 17, 21cbvralw 3420 . . . . . . . 8 (∀𝑙 ∈ (ℤ𝑗)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))
238, 22syl6bb 289 . . . . . . 7 (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
2423cbvrexvw 3429 . . . . . 6 (∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑢) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))
256, 24syl6bb 289 . . . . 5 (𝑣 = 𝑢 → (∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
263, 25imbi12d 347 . . . 4 (𝑣 = 𝑢 → ((𝑃𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣)) ↔ (𝑃𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢))))
2726cbvralvw 3428 . . 3 (∀𝑣𝐽 (𝑃𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹𝑙) ∈ 𝑣)) ↔ ∀𝑢𝐽 (𝑃𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑢)))
28273anbi3i 1155 . 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 289 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 208  wa 398  w3a 1083  wcel 2114  wnfc 2957  wral 3125  wrex 3126   class class class wbr 5042  dom cdm 5531  cfv 6331  (class class class)co 7133  pm cpm 8385  cc 10513  cz 11960  cuz 12222  TopOnctopon 21494  𝑡clm 21810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-addrcl 10576  ax-rnegex 10586  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-po 5450  df-so 5451  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-er 8267  df-pm 8387  df-en 8488  df-dom 8489  df-sdom 8490  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-neg 10851  df-z 11961  df-uz 12223  df-top 21478  df-topon 21495  df-lm 21813
This theorem is referenced by:  xlimbr  42260  xlimmnfvlem1  42265  xlimmnfvlem2  42266  xlimpnfvlem1  42269  xlimpnfvlem2  42270
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