Theorem fbflim2 22579

Description: A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 22578. (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 22578	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))))
3		topontop 21515	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4	3	ad2antrr 724	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
5		simpr 487	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
6		toponuni 21516	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
8	5, 7	eleqtrd 2915	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐽)
9		eqid 2821	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	isneip 21707	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))))
11	4, 8, 10	syl2anc 586	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))))
12		simpr 487	. . . . . . 7 ⊢ ((𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))
13	11, 12	syl6bi 255	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)))
14		r19.29 3254	. . . . . . . 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		pm3.45 623	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) → ((𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛)))
16	15	imp 409	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛))
17		sstr2 3973	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑛 → 𝑥 ⊆ 𝑛))
18	17	com12 32	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ 𝑛 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ 𝑛))
19	18	reximdv 3273	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑛 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))
20	19	impcom 410	. . . . . . . . . 10 ⊢ ((∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)
21	16, 20	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)
22	21	rexlimivw 3282	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐽 ((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)
23	14, 22	syl 17	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛)
24	23	ex 415	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) → (∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛) → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))
25	13, 24	syl9 77	. . . . 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 3188	. . . 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 483	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝐽 ∈ Top)
28		simprl 769	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝑦 ∈ 𝐽)
29		simprr 771	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
30		opnneip 21721	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
31	27, 28, 29, 30	syl3anc 1367	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
32		sseq2 3992	. . . . . . . . . 10 ⊢ (𝑛 = 𝑦 → (𝑥 ⊆ 𝑛 ↔ 𝑥 ⊆ 𝑦))
33	32	rexbidv 3297	. . . . . . . . 9 ⊢ (𝑛 = 𝑦 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 ↔ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))
34	33	rspcv 3617	. . . . . . . 8 ⊢ (𝑦 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))
35	31, 34	syl 17	. . . . . . 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 459	. . . . . 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 86	. . . . 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 3189	. . . 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 214	. . 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 581	. 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 281	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 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  {csn 4560  ∪ cuni 4831  ‘cfv 6349  (class class class)co 7150  fBascfbas 20527  filGencfg 20528  Topctop 21495  TopOnctopon 21512  neicnei 21699   fLim cflim 22536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-fbas 20536  df-fg 20537  df-top 21496  df-topon 21513  df-ntr 21622  df-nei 21700  df-fil 22448  df-flim 22541

This theorem is referenced by: (None)