Theorem flimclsi 23986

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 2737	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	flimfil 23977	. . . . . . 7 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
3	2	ad2antlr 727	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘∪ 𝐽))
4		flimnei 23975	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹)
5	4	adantll 714	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹)
6		simpll 767	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ∈ 𝐹)
7		filinn0 23868	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑦 ∈ 𝐹 ∧ 𝑆 ∈ 𝐹) → (𝑦 ∩ 𝑆) ≠ ∅)
8	3, 5, 6, 7	syl3anc 1373	. . . . 5 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦 ∩ 𝑆) ≠ ∅)
9	8	ralrimiva 3146	. . . 4 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅)
10		flimtop 23973	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
11	10	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top)
12		filelss 23860	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑆 ∈ 𝐹) → 𝑆 ⊆ ∪ 𝐽)
13	12	ancoms 458	. . . . . 6 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → 𝑆 ⊆ ∪ 𝐽)
14	2, 13	sylan2 593	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑆 ⊆ ∪ 𝐽)
15	1	flimelbas 23976	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ∪ 𝐽)
16	15	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ∪ 𝐽)
17	1	neindisj2 23131	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅))
18	11, 14, 16, 17	syl3anc 1373	. . . 4 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅))
19	9, 18	mpbird 257	. . 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 3989	1 ⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626  ∪ cuni 4907  ‘cfv 6561  (class class class)co 7431  Topctop 22899  clsccl 23026  neicnei 23105  Filcfil 23853   fLim cflim 23942

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-top 22900  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-fil 23854  df-flim 23947

This theorem is referenced by:  flimcls  23993  flimfcls  24034  cnextcn  24075  cmetss  25350  minveclem4  25466