Theorem flimclsi 24007

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 2740	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	flimfil 23998	. . . . . . 7 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
3	2	ad2antlr 726	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝐹 ∈ (Fil‘∪ 𝐽))
4		flimnei 23996	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹)
5	4	adantll 713	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑦 ∈ 𝐹)
6		simpll 766	. . . . . 6 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ∈ 𝐹)
7		filinn0 23889	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐽) ∧ 𝑦 ∈ 𝐹 ∧ 𝑆 ∈ 𝐹) → (𝑦 ∩ 𝑆) ≠ ∅)
8	3, 5, 6, 7	syl3anc 1371	. . . . 5 ⊢ (((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ∧ 𝑦 ∈ ((nei‘𝐽)‘{𝑥})) → (𝑦 ∩ 𝑆) ≠ ∅)
9	8	ralrimiva 3152	. . . 4 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅)
10		flimtop 23994	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
11	10	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝐽 ∈ Top)
12		filelss 23881	. . . . . . 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 23997	. . . . . 6 ⊢ (𝑥 ∈ (𝐽 fLim 𝐹) → 𝑥 ∈ ∪ 𝐽)
16	15	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ 𝐹 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) → 𝑥 ∈ ∪ 𝐽)
17	1	neindisj2 23152	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑦 ∈ ((nei‘𝐽)‘{𝑥})(𝑦 ∩ 𝑆) ≠ ∅))
18	11, 14, 16, 17	syl3anc 1371	. . . 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 4014	1 ⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∪ cuni 4931  ‘cfv 6573  (class class class)co 7448  Topctop 22920  clsccl 23047  neicnei 23126  Filcfil 23874   fLim cflim 23963

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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-fbas 21384  df-top 22921  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-fil 23875  df-flim 23968

This theorem is referenced by:  flimcls  24014  flimfcls  24055  cnextcn  24096  cmetss  25369  minveclem4  25485