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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  Ⅎwnfc 2957  ∀wral 3125  ∃wrex 3126   class class class wbr 5042  dom cdm 5531  ‘cfv 6331  (class class class)co 7133   ↑pm cpm 8385  ℂcc 10513  ℤcz 11960  ℤ≥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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