Theorem neifil 24192

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 23225	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
2	1	adantr 486	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ 𝐽)
3		topontop 23224	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4	3	adantr 486	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
5		simpr 490	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
6	5, 2	sseqtrd 3967	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ∪ 𝐽)
7		eqid 2761	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
8	7	neiuni 23433	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ ((nei‘𝐽)‘𝑆))
9	4, 6, 8	syl2anc 596	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∪ 𝐽 = ∪ ((nei‘𝐽)‘𝑆))
10	2, 9	eqtrd 2796	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆))
11		eqimss2 3990	. . . . . 6 ⊢ (𝑋 = ∪ ((nei‘𝐽)‘𝑆) → ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
12	10, 11	syl 18	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
13		sspwuni 5060	. . . . 5 ⊢ (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ↔ ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
14	12, 13	sylibr 237	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
15	14	3adant3 1150	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
16		0nnei 23423	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
17	3, 16	sylan 592	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
18	17	3adant2 1149	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
19	7	tpnei 23432	. . . . . . 7 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ ∪ 𝐽 ↔ ∪ 𝐽 ∈ ((nei‘𝐽)‘𝑆)))
20	19	biimpa 482	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ∪ 𝐽 ∈ ((nei‘𝐽)‘𝑆))
21	4, 6, 20	syl2anc 596	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∪ 𝐽 ∈ ((nei‘𝐽)‘𝑆))
22	2, 21	eqeltrd 2861	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
23	22	3adant3 1150	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
24	15, 18, 23	3jca 1146	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
25		elpwi 4564	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
26	4	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝐽 ∈ Top)
27		simprl 783	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ ((nei‘𝐽)‘𝑆))
28		simprr 785	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
29		simplr 781	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ 𝑋)
30	2	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑋 = ∪ 𝐽)
31	29, 30	sseqtrd 3967	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
32	7	ssnei2 23427	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ ∪ 𝐽)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
33	26, 27, 28, 31, 32	syl22anc 852	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
34	33	rexlimdvaa 3165	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
35	25, 34	sylan2 605	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
36	35	ralrimiva 3155	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
37	36	3adant3 1150	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
38		innei 23436	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆))
39	38	3expib 1140	. . . . 5 ⊢ (𝐽 ∈ Top → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆)))
40	3, 39	syl 18	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆)))
41	40	3ad2ant1 1151	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆)))
42	41	ralrimivv 3204	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆))
43		isfil2 24168	. 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 1362	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867  ‘cfv 6537  Topctop 23204  TopOnctopon 23221  neicnei 23408  Filcfil 24157

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-fbas 21668  df-top 23205  df-topon 23222  df-nei 23409  df-fil 24158

This theorem is used by:  trnei  24204  neiflim  24286  hausflim  24293  flimcf  24294  flimclslem  24296  cnpflf2  24312  cnpflf  24313  fclsfnflim  24339  neipcfilu  24607