Theorem fbflim2 24115

Description: A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 24114. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
fbflim.3	⊢ 𝐹 = (𝑋filGen𝐵)

Assertion

Ref	Expression
fbflim2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)))

Distinct variable groups:   𝑥,𝑛,𝐴   𝐵,𝑛,𝑥   𝑛,𝐽,𝑥   𝑛,𝑋,𝑥   𝑥,𝐹

Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem fbflim2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fbflim.3	. . 3 ⊢ 𝐹 = (𝑋filGen𝐵)
2	1	fbflim 24114	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))))
3		topontop 23051	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4	3	ad2antrr 738	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
5		simpr 489	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
6		toponuni 23052	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
8	5, 7	eleqtrd 2865	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐽)
9		eqid 2763	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	isneip 23243	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))))
11	4, 8, 10	syl2anc 595	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))))
12		simpr 489	. . . . . . 7 ⊢ ((𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))
13	11, 12	biimtrdi 256	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)))
14		r19.29 3128	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑦 ∈ 𝐽 ((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)))
15		pm3.45 633	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) → ((𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛)))
16	15	imp 411	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛))
17		sstr2 3945	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑛 → 𝑥 ⊆ 𝑛))
18	17	com12 33	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ 𝑛 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ 𝑛))
19	18	reximdv 3180	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑛 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))
20	19	impcom 412	. . . . . . . . . 10 ⊢ ((∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)
21	16, 20	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)
22	21	rexlimivw 3162	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐽 ((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)
23	14, 22	syl 18	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)
24	23	ex 417	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) → (∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))
25	13, 24	syl9 78	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)))
26	25	ralrimdv 3163	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))
27	4	adantr 485	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝐽 ∈ Top)
28		simprl 782	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝑦 ∈ 𝐽)
29		simprr 784	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
30		opnneip 23257	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
31	27, 28, 29, 30	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
32		sseq2 3964	. . . . . . . . . 10 ⊢ (𝑛 = 𝑦 → (𝑥 ⊆ 𝑛 ↔ 𝑥 ⊆ 𝑦))
33	32	rexbidv 3189	. . . . . . . . 9 ⊢ (𝑛 = 𝑦 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 ↔ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))
34	33	rspcv 3578	. . . . . . . 8 ⊢ (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))
35	31, 34	syl 18	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))
36	35	expr 461	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦)))
37	36	com23 87	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐽) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 → (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦)))
38	37	ralrimdva 3165	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 → ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦)))
39	26, 38	impbid 215	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))
40	39	pm5.32da 589	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)))
41	2, 40	bitrd 282	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3906  {csn 4590  ∪ cuni 4873  ‘cfv 6538  (class class class)co 7412  fBascfbas 21491  filGencfg 21492  Topctop 23031  TopOnctopon 23048  neicnei 23235   fLim cflim 24072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-fbas 21500  df-fg 21501  df-top 23032  df-topon 23049  df-ntr 23158  df-nei 23236  df-fil 23984  df-flim 24077

This theorem is referenced by: (None)