Theorem lmbr3 44463

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 44461	. 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 2818	. . . . 5 ⊢ (𝑣 = 𝑢 → (𝑃 ∈ 𝑣 ↔ 𝑃 ∈ 𝑢))
4		eleq2w 2818	. . . . . . . 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 3221	. . . . . 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 6892	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗))
8	7	raleqdv 3326	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢)))
9		nfcv 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑙
10		lmbr3.1	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝐹
11	10	nfdm 5951	. . . . . . . . . . 11 ⊢ Ⅎ𝑘dom 𝐹
12	9, 11	nfel 2918	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑙 ∈ dom 𝐹
13	10, 9	nffv 6902	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐹‘𝑙)
14		nfcv 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑢
15	13, 14	nfel 2918	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝐹‘𝑙) ∈ 𝑢
16	12, 15	nfan 1903	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑙 ∈ dom 𝐹 ∧ (𝐹‘𝑙) ∈ 𝑢)
17		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑙(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)
18		eleq1w 2817	. . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (𝑙 ∈ dom 𝐹 ↔ 𝑘 ∈ dom 𝐹))
19		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘))
20	19	eleq1d 2819	. . . . . . . . . 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 3304	. . . . . . . 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 3236	. . . . . 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 345	. . . 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 3235	. . 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 205   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107  Ⅎwnfc 2884  ∀wral 3062  ∃wrex 3071   class class class wbr 5149  dom cdm 5677  ‘cfv 6544  (class class class)co 7409   ↑_pm cpm 8821  ℂcc 11108  ℤcz 12558  ℤ_≥cuz 12822  TopOnctopon 22412  ⇝_𝑡clm 22730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-addrcl 11171  ax-rnegex 11181  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-er 8703  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-neg 11447  df-z 12559  df-uz 12823  df-top 22396  df-topon 22413  df-lm 22733

This theorem is referenced by:  xlimbr  44543  xlimmnfvlem1  44548  xlimmnfvlem2  44549  xlimpnfvlem1  44552  xlimpnfvlem2  44553