Theorem fbflim2 23967

Description: A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 23966. (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 23966	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))))
3		topontop 22903	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4	3	ad2antrr 732	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
5		simpr 485	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
6		toponuni 22904	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
7	6	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝑋 = ∪ 𝐽)
8	5, 7	eleqtrd 2842	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐽)
9		eqid 2740	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	isneip 23095	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))))
11	4, 8, 10	syl2anc 590	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))))
12		simpr 485	. . . . . . 7 ⊢ ((𝑛 ⊆ ∪ 𝐽 ∧ ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛))
13	11, 12	biimtrdi 254	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)))
14		r19.29 3103	. . . . . . . 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		pm3.45 628	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) → ((𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛)))
16	15	imp 407	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦) ∧ (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑛)) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑛))
17		sstr2 3929	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑦 → (𝑦 ⊆ 𝑛 → 𝑥 ⊆ 𝑛))
18	17	com12 32	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ 𝑛 → (𝑥 ⊆ 𝑦 → 𝑥 ⊆ 𝑛))
19	18	reximdv 3155	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑛 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))
20	19	impcom 408	. . . . . . . . . 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 3137	. . . . . . . 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 413	. . . . . 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 3138	. . . 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 481	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝐽 ∈ Top)
28		simprl 776	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝑦 ∈ 𝐽)
29		simprr 778	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝐴 ∈ 𝑦)
30		opnneip 23109	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
31	27, 28, 29, 30	syl3anc 1379	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐽 ∧ 𝐴 ∈ 𝑦)) → 𝑦 ∈ ((nei‘𝐽)‘{𝐴}))
32		sseq2 3948	. . . . . . . . . 10 ⊢ (𝑛 = 𝑦 → (𝑥 ⊆ 𝑛 ↔ 𝑥 ⊆ 𝑦))
33	32	rexbidv 3164	. . . . . . . . 9 ⊢ (𝑛 = 𝑦 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛 ↔ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑦))
34	33	rspcv 3563	. . . . . . . 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 457	. . . . . 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 3140	. . . 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 213	. . 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 584	. 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 280	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 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064   ⊆ wss 3890  {csn 4562  ∪ cuni 4845  ‘cfv 6492  (class class class)co 7363  fBascfbas 21342  filGencfg 21343  Topctop 22883  TopOnctopon 22900  neicnei 23087   fLim cflim 23924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-fbas 21351  df-fg 21352  df-top 22884  df-topon 22901  df-ntr 23010  df-nei 23088  df-fil 23836  df-flim 23929

This theorem is referenced by: (None)