Theorem flimclsi 23157

Description: The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
flimclsi	⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Proof of Theorem flimclsi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	flimfil 23148	. . . . . . 7 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
3	2	ad2antlr 723	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘∪ 𝐽))
4		flimnei 23146	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹)
5	4	adantll 710	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹)
6		simpll 763	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ∈ 𝐹)
7		filinn0 23039	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑦 ∈ 𝐹 ∧ 𝑆 ∈ 𝐹) → (𝑦 ∩ 𝑆) ≠ ∅)
8	3, 5, 6, 7	syl3anc 1369	. . . . 5 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦 ∩ 𝑆) ≠ ∅)
9	8	ralrimiva 3137	. . . 4 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅)
10		flimtop 23144	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
11	10	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top)
12		filelss 23031	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑆 ∈ 𝐹) → 𝑆 ⊆ ∪ 𝐽)
13	12	ancoms 458	. . . . . 6 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → 𝑆 ⊆ ∪ 𝐽)
14	2, 13	sylan2 592	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑆 ⊆ ∪ 𝐽)
15	1	flimelbas 23147	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ∪ 𝐽)
16	15	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ∪ 𝐽)
17	1	neindisj2 22302	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅))
18	11, 14, 16, 17	syl3anc 1369	. . . 4 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅))
19	9, 18	mpbird 256	. . 3 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝑆))
20	19	ex 412	. 2 ⊢ (𝑆 ∈ 𝐹 → (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘𝑆)))
21	20	ssrdv 3929	1 ⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2101   ≠ wne 2938  ∀wral 3059   ∩ cin 3888   ⊆ wss 3889  ∅c0 4259  {csn 4564  ∪ cuni 4841  ‘cfv 6447  (class class class)co 7295  Topctop 22070  clsccl 22197  neicnei 22276  Filcfil 23024   fLim cflim 23113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-int 4883  df-iun 4929  df-iin 4930  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-fbas 20622  df-top 22071  df-cld 22198  df-ntr 22199  df-cls 22200  df-nei 22277  df-fil 23025  df-flim 23118

This theorem is referenced by:  flimcls  23164  flimfcls  23205  cnextcn  23246  cmetss  24508  minveclem4  24624