Theorem lmbr3 45728

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	⊢ Ⅎ𝑘𝐹
lmbr3.2	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))

Assertion

Ref	Expression
lmbr3	⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 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.

Step	Hyp	Ref	Expression
1		lmbr3.2	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2	1	lmbr3v 45726	. 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑣)))))
3		eleq2w 2812	. . . . 5 ⊢ (𝑣 = 𝑢 → (𝑃 ∈ 𝑣 ↔ 𝑃 ∈ 𝑢))
4		eleq2w 2812	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝑙) ∈ 𝑣 ↔ (𝐹‘𝑙) ∈ 𝑢))
5	4	anbi2d 630	. . . . . . 7 ⊢ (𝑣 = 𝑢 → ((𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑣) ↔ (𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢)))
6	5	rexralbidv 3195	. . . . . 6 ⊢ (𝑣 = 𝑢 → (∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑣) ↔ ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢)))
7		fveq2 6822	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗))
8	7	raleqdv 3289	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢)))
9		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑙
10		lmbr3.1	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝐹
11	10	nfdm 5893	. . . . . . . . . . 11 ⊢ Ⅎ𝑘dom 𝐹
12	9, 11	nfel 2906	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑙 ∈ dom 𝐹
13	10, 9	nffv 6832	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐹‘𝑙)
14		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑢
15	13, 14	nfel 2906	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝐹‘𝑙) ∈ 𝑢
16	12, 15	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢)
17		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑙(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)
18		eleq1w 2811	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (𝑙 ∈ dom 𝐹 ↔ 𝑘 ∈ dom 𝐹))
19		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘))
20	19	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ 𝑢))
21	18, 20	anbi12d 632	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → ((𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
22	16, 17, 21	cbvralw 3271	. . . . . . . 8 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑗)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
23	8, 22	bitrdi 287	. . . . . . 7 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
24	23	cbvrexvw 3208	. . . . . 6 ⊢ (∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))
25	6, 24	bitrdi 287	. . . . 5 ⊢ (𝑣 = 𝑢 → (∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑣) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
26	3, 25	imbi12d 344	. . . 4 ⊢ (𝑣 = 𝑢 → ((𝑃 ∈ 𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑣)) ↔ (𝑃 ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))
27	26	cbvralvw 3207	. . 3 ⊢ (∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑣)) ↔ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
28	27	3anbi3i 1159	. 2 ⊢ ((𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑣 ∈ 𝐽 (𝑃 ∈ 𝑣 → ∃𝑖 ∈ ℤ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑣))) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))
29	2, 28	bitrdi 287	1 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  Ⅎwnfc 2876  ∀wral 3044  ∃wrex 3053   class class class wbr 5092  dom cdm 5619  ‘cfv 6482  (class class class)co 7349   ↑_pm cpm 8754  ℂcc 11007  ℤcz 12471  ℤ_≥cuz 12735  TopOnctopon 22795  ⇝_𝑡clm 23111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-addrcl 11070  ax-rnegex 11080  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-er 8625  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-neg 11350  df-z 12472  df-uz 12736  df-top 22779  df-topon 22796  df-lm 23114

This theorem is referenced by:  xlimbr  45808  xlimmnfvlem1  45813  xlimmnfvlem2  45814  xlimpnfvlem1  45817  xlimpnfvlem2  45818