Theorem neifil 23824

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 22858	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ 𝐽)
3		topontop 22857	. . . . . . . . 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 3970	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ∪ 𝐽)
7		eqid 2736	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
8	7	neiuni 23066	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ∪ 𝐽 = ∪ ((nei‘𝐽)‘𝑆))
9	4, 6, 8	syl2anc 584	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∪ 𝐽 = ∪ ((nei‘𝐽)‘𝑆))
10	2, 9	eqtrd 2771	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑋 = ∪ ((nei‘𝐽)‘𝑆))
11		eqimss2 3993	. . . . . 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 5055	. . . . 5 ⊢ (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ↔ ∪ ((nei‘𝐽)‘𝑆) ⊆ 𝑋)
14	12, 13	sylibr 234	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
15	14	3adant3 1132	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋)
16		0nnei 23056	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
17	3, 16	sylan 580	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
18	17	3adant2 1131	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ¬ ∅ ∈ ((nei‘𝐽)‘𝑆))
19	7	tpnei 23065	. . . . . . 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 584	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∪ 𝐽 ∈ ((nei‘𝐽)‘𝑆))
22	2, 21	eqeltrd 2836	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
23	22	3adant3 1132	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → 𝑋 ∈ ((nei‘𝐽)‘𝑆))
24	15, 18, 23	3jca 1128	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (((nei‘𝐽)‘𝑆) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)))
25		elpwi 4561	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
26	4	ad2antrr 726	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝐽 ∈ Top)
27		simprl 770	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ ((nei‘𝐽)‘𝑆))
28		simprr 772	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
29		simplr 768	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ 𝑋)
30	2	ad2antrr 726	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑋 = ∪ 𝐽)
31	29, 30	sseqtrd 3970	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
32	7	ssnei2 23060	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ ∪ 𝐽)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
33	26, 27, 28, 31, 32	syl22anc 838	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) ∧ (𝑦 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
34	33	rexlimdvaa 3138	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
35	25, 34	sylan2 593	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
36	35	ralrimiva 3128	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
37	36	3adant3 1132	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝑦 ⊆ 𝑥 → 𝑥 ∈ ((nei‘𝐽)‘𝑆)))
38		innei 23069	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆))
39	38	3expib 1122	. . . . 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 1133	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑦 ∈ ((nei‘𝐽)‘𝑆)) → (𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆)))
42	41	ralrimivv 3177	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∀𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦 ∈ ((nei‘𝐽)‘𝑆)(𝑥 ∩ 𝑦) ∈ ((nei‘𝐽)‘𝑆))
43		isfil2 23800	. 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 1344	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863  ‘cfv 6492  Topctop 22837  TopOnctopon 22854  neicnei 23041  Filcfil 23789

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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-fbas 21306  df-top 22838  df-topon 22855  df-nei 23042  df-fil 23790

This theorem is referenced by:  trnei  23836  neiflim  23918  hausflim  23925  flimcf  23926  flimclslem  23928  cnpflf2  23944  cnpflf  23945  fclsfnflim  23971  neipcfilu  24239