Theorem trnei 23867

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 23862 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 22888	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑌) → 𝐽 ∈ Top)
2	1	3ad2ant1 1134	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐽 ∈ Top)
3		simp2 1138	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐴 ⊆ 𝑌)
4		toponuni 22889	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐽)
5	4	3ad2ant1 1134	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑌 = ∪ 𝐽)
6	3, 5	sseqtrd 3959	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐴 ⊆ ∪ 𝐽)
7		simp3 1139	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑃 ∈ 𝑌)
8	7, 5	eleqtrd 2839	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑃 ∈ ∪ 𝐽)
9		eqid 2737	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	neindisj2 23098	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅))
11	2, 6, 8, 10	syl3anc 1374	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅))
12		simp1 1137	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑌))
13	7	snssd 4753	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ⊆ 𝑌)
14		snnzg 4719	. . . . 5 ⊢ (𝑃 ∈ 𝑌 → {𝑃} ≠ ∅)
15	14	3ad2ant3 1136	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ≠ ∅)
16		neifil 23855	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
17	12, 13, 15, 16	syl3anc 1374	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
18		trfil2 23862	. . 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 585	. 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 282	1 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ cuni 4851  ‘cfv 6492  (class class class)co 7360   ↾_t crest 17374  Topctop 22868  TopOnctopon 22885  clsccl 22993  neicnei 23072  Filcfil 23820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  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-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-rest 17376  df-fbas 21341  df-top 22869  df-topon 22886  df-cld 22994  df-ntr 22995  df-cls 22996  df-nei 23073  df-fil 23821

This theorem is referenced by:  flfcntr  24018  cnextfun  24039  cnextfvval  24040  cnextf  24041  cnextcn  24042  cnextfres1  24043  cnextucn  24277  ucnextcn  24278  limcflflem  25857  rrhre  34181