Theorem fbflim2 23714

Description: A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 23713. (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 23713	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))))
3		topontop 22648	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4	3	ad2antrr 723	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
5		simpr 484	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
6		toponuni 22649	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
8	5, 7	eleqtrd 2834	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐽)
9		eqid 2731	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	isneip 22842	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))))
11	4, 8, 10	syl2anc 583	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))))
12		simpr 484	. . . . . . 7 ⊢ ((𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))
13	11, 12	syl6bi 253	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)))
14		r19.29 3113	. . . . . . . 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		pm3.45 621	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) → ((𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛)))
16	15	imp 406	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛))
17		sstr2 3989	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑛 → 𝑥 ⊆ 𝑛))
18	17	com12 32	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ 𝑛 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ 𝑛))
19	18	reximdv 3169	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑛 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))
20	19	impcom 407	. . . . . . . . . 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 3150	. . . . . . . 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 412	. . . . . 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 3151	. . . 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 480	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝐽 ∈ Top)
28		simprl 768	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝑦 ∈ 𝐽)
29		simprr 770	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
30		opnneip 22856	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
31	27, 28, 29, 30	syl3anc 1370	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
32		sseq2 4008	. . . . . . . . . 10 ⊢ (𝑛 = 𝑦 → (𝑥 ⊆ 𝑛 ↔ 𝑥 ⊆ 𝑦))
33	32	rexbidv 3177	. . . . . . . . 9 ⊢ (𝑛 = 𝑦 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 ↔ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))
34	33	rspcv 3608	. . . . . . . 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 456	. . . . . 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 3153	. . . 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 211	. . 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 578	. 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 279	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 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069   ⊆ wss 3948  {csn 4628  ∪ cuni 4908  ‘cfv 6543  (class class class)co 7412  fBascfbas 21136  filGencfg 21137  Topctop 22628  TopOnctopon 22645  neicnei 22834   fLim cflim 23671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-fbas 21145  df-fg 21146  df-top 22629  df-topon 22646  df-ntr 22757  df-nei 22835  df-fil 23583  df-flim 23676

This theorem is referenced by: (None)