Theorem lmbr3 46175

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 46173	. 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 2820	. . . . 5 ⊢ (𝑣 = 𝑢 → (𝑃 ∈ 𝑣 ↔ 𝑃 ∈ 𝑢))
4		eleq2w 2820	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝑙) ∈ 𝑣 ↔ (𝐹‘𝑙) ∈ 𝑢))
5	4	anbi2d 631	. . . . . . 7 ⊢ (𝑣 = 𝑢 → ((𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑣) ↔ (𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢)))
6	5	rexralbidv 3203	. . . . . 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 6840	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗))
8	7	raleqdv 3295	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢)))
9		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑙
10		lmbr3.1	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝐹
11	10	nfdm 5906	. . . . . . . . . . 11 ⊢ Ⅎ𝑘dom 𝐹
12	9, 11	nfel 2913	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑙 ∈ dom 𝐹
13	10, 9	nffv 6850	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐹‘𝑙)
14		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑢
15	13, 14	nfel 2913	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝐹‘𝑙) ∈ 𝑢
16	12, 15	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢)
17		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑙(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)
18		eleq1w 2819	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (𝑙 ∈ dom 𝐹 ↔ 𝑘 ∈ dom 𝐹))
19		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘))
20	19	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ 𝑢))
21	18, 20	anbi12d 633	. . . . . . . . 9 ⊢ (𝑙 = 𝑘 → ((𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))
22	16, 17, 21	cbvralw 3279	. . . . . . . 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 3216	. . . . . 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 3215	. . 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 1160	. 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 1087   ∈ wcel 2114  Ⅎwnfc 2883  ∀wral 3051  ∃wrex 3061   class class class wbr 5085  dom cdm 5631  ‘cfv 6498  (class class class)co 7367   ↑_pm cpm 8774  ℂcc 11036  ℤcz 12524  ℤ_≥cuz 12788  TopOnctopon 22875  ⇝_𝑡clm 23191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-addrcl 11099  ax-rnegex 11109  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-er 8643  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-neg 11380  df-z 12525  df-uz 12789  df-top 22859  df-topon 22876  df-lm 23194

This theorem is referenced by:  xlimbr  46255  xlimmnfvlem1  46260  xlimmnfvlem2  46261  xlimpnfvlem1  46264  xlimpnfvlem2  46265