Theorem trnei 24266

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 24261 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 23286	. . . 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 23287	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐽)
5	4	3ad2ant1 1151	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑌 = ∪ 𝐽)
6	3, 5	sseqtrd 3949	. . 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 2866	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑃 ∈ ∪ 𝐽)
9		eqid 2763	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	neindisj2 23496	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅))
11	2, 6, 8, 10	syl3anc 1399	. 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 4749	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ⊆ 𝑌)
14		snnzg 4737	. . . . 5 ⊢ (𝑃 ∈ 𝑌 → {𝑃} ≠ ∅)
15	14	3ad2ant3 1153	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ≠ ∅)
16		neifil 24254	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
17	12, 13, 15, 16	syl3anc 1399	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
18		trfil2 24261	. . 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 2147   ≠ wne 2959  ∀wral 3080   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  {csn 4588  ∪ cuni 4872  ‘cfv 6566  (class class class)co 7451   ↾_t crest 17641  Topctop 23266  TopOnctopon 23283  clsccl 23391  neicnei 23470  Filcfil 24219

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-rest 17643  df-fbas 21726  df-top 23267  df-topon 23284  df-cld 23392  df-ntr 23393  df-cls 23394  df-nei 23471  df-fil 24220

This theorem is used by:  flfcntr  24417  cnextfun  24438  cnextfvval  24439  cnextf  24440  cnextcn  24441  cnextfres1  24442  cnextucn  24676  ucnextcn  24677  limcflflem  26255  rrhre  34764