Theorem neifil 22939

Description: The neighborhoods of a nonempty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
neifil	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))

Proof of Theorem neifil

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		toponuni 21971	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ 𝐽)
3		topontop 21970	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
5		simpr 484	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
6	5, 2	sseqtrd 3957	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ∪ 𝐽)
7		eqid 2738	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
8	7	neiuni 22181	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ ((nei‘𝐽)‘𝑆))
9	4, 6, 8	syl2anc 583	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∪ 𝐽 = ∪ ((nei‘𝐽)‘𝑆))
10	2, 9	eqtrd 2778	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆))
11		eqimss2 3974	. . . . . 6 ⊢ (𝑋 = ∪ ((nei‘𝐽)‘𝑆) → ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
12	10, 11	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
13		sspwuni 5025	. . . . 5 ⊢ (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ↔ ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
14	12, 13	sylibr 233	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
15	14	3adant3 1130	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
16		0nnei 22171	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
17	3, 16	sylan 579	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
18	17	3adant2 1129	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
19	7	tpnei 22180	. . . . . . 7 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ ∪ 𝐽 ↔ ∪ 𝐽 ∈ ((nei‘𝐽)‘𝑆)))
20	19	biimpa 476	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘𝑆))
21	4, 6, 20	syl2anc 583	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∪ 𝐽 ∈ ((nei‘𝐽)‘𝑆))
22	2, 21	eqeltrd 2839	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
23	22	3adant3 1130	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
24	15, 18, 23	3jca 1126	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
25		elpwi 4539	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
26	4	ad2antrr 722	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝐽 ∈ Top)
27		simprl 767	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ ((nei‘𝐽)‘𝑆))
28		simprr 769	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
29		simplr 765	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ 𝑋)
30	2	ad2antrr 722	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑋 = ∪ 𝐽)
31	29, 30	sseqtrd 3957	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
32	7	ssnei2 22175	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ ∪ 𝐽)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
33	26, 27, 28, 31, 32	syl22anc 835	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
34	33	rexlimdvaa 3213	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
35	25, 34	sylan2 592	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
36	35	ralrimiva 3107	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
37	36	3adant3 1130	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
38		innei 22184	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆))
39	38	3expib 1120	. . . . 5 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆)))
40	3, 39	syl 17	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆)))
41	40	3ad2ant1 1131	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆)))
42	41	ralrimivv 3113	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆))
43		isfil2 22915	. 2 ⊢ (((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋) ↔ ((((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)) ∧ ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆)))
44	24, 37, 42, 43	syl3anbrc 1341	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  TopOnctopon 21967  neicnei 22156  Filcfil 22904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-fbas 20507  df-top 21951  df-topon 21968  df-nei 22157  df-fil 22905

This theorem is referenced by:  trnei  22951  neiflim  23033  hausflim  23040  flimcf  23041  flimclslem  23043  cnpflf2  23059  cnpflf  23060  fclsfnflim  23086  neipcfilu  23356