Theorem trnei 24078

Description: The trace, over a set 𝐴, of the filter of the neighborhoods of a point 𝑃 is a filter iff 𝑃 belongs to the closure of 𝐴. (This is trfil2 24073 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
trnei	⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴)))

Proof of Theorem trnei

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 23099	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑌) → 𝐽 ∈ Top)
2	1	3ad2ant1 1151	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐽 ∈ Top)
3		simp2 1155	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐴 ⊆ 𝑌)
4		toponuni 23100	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐽)
5	4	3ad2ant1 1151	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑌 = ∪ 𝐽)
6	3, 5	sseqtrd 3974	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐴 ⊆ ∪ 𝐽)
7		simp3 1156	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑃 ∈ 𝑌)
8	7, 5	eleqtrd 2867	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑃 ∈ ∪ 𝐽)
9		eqid 2765	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	neindisj2 23309	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅))
11	2, 6, 8, 10	syl3anc 1398	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅))
12		simp1 1154	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑌))
13	7	snssd 4754	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ⊆ 𝑌)
14		snnzg 4742	. . . . 5 ⊢ (𝑃 ∈ 𝑌 → {𝑃} ≠ ∅)
15	14	3ad2ant3 1153	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ≠ ∅)
16		neifil 24066	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
17	12, 13, 15, 16	syl3anc 1398	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
18		trfil2 24073	. . 3 ⊢ ((((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅))
19	17, 3, 18	syl2anc 596	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅))
20	11, 19	bitr4d 285	1 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591  ∪ cuni 4874  ‘cfv 6540  (class class class)co 7416   ↾_t crest 17490  Topctop 23079  TopOnctopon 23096  clsccl 23204  neicnei 23283  Filcfil 24031

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-rest 17492  df-fbas 21548  df-top 23080  df-topon 23097  df-cld 23205  df-ntr 23206  df-cls 23207  df-nei 23284  df-fil 24032

This theorem is used by:  flfcntr  24229  cnextfun  24250  cnextfvval  24251  cnextf  24252  cnextcn  24253  cnextfres1  24254  cnextucn  24488  ucnextcn  24489  limcflflem  26068  rrhre  34434