Theorem trnei 23779

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 23774 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 22800	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑌) → 𝐽 ∈ Top)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐽 ∈ Top)
3		simp2 1137	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐴 ⊆ 𝑌)
4		toponuni 22801	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐽)
5	4	3ad2ant1 1133	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑌 = ∪ 𝐽)
6	3, 5	sseqtrd 3983	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐴 ⊆ ∪ 𝐽)
7		simp3 1138	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑃 ∈ 𝑌)
8	7, 5	eleqtrd 2830	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑃 ∈ ∪ 𝐽)
9		eqid 2729	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	neindisj2 23010	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅))
11	2, 6, 8, 10	syl3anc 1373	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅))
12		simp1 1136	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑌))
13	7	snssd 4773	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ⊆ 𝑌)
14		snnzg 4738	. . . . 5 ⊢ (𝑃 ∈ 𝑌 → {𝑃} ≠ ∅)
15	14	3ad2ant3 1135	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ≠ ∅)
16		neifil 23767	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
17	12, 13, 15, 16	syl3anc 1373	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌))
18		trfil2 23774	. . 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 584	. 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 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  {csn 4589  ∪ cuni 4871  ‘cfv 6511  (class class class)co 7387   ↾_t crest 17383  Topctop 22780  TopOnctopon 22797  clsccl 22905  neicnei 22984  Filcfil 23732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-rest 17385  df-fbas 21261  df-top 22781  df-topon 22798  df-cld 22906  df-ntr 22907  df-cls 22908  df-nei 22985  df-fil 23733

This theorem is referenced by:  flfcntr  23930  cnextfun  23951  cnextfvval  23952  cnextf  23953  cnextcn  23954  cnextfres1  23955  cnextucn  24190  ucnextcn  24191  limcflflem  25781  rrhre  34011