Theorem flimfcls 23924

Description: A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
flimfcls	⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)

Proof of Theorem flimfcls

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

Step	Hyp	Ref	Expression
1		flimtop 23863	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2		eqid 2728	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	flimfil 23867	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
4		flimclsi 23876	. . . . . 6 ⊢ (𝑥 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥))
5	4	sseld 3978	. . . . 5 ⊢ (𝑥 ∈ 𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
6	5	com12 32	. . . 4 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥 ∈ 𝐹 → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
7	6	ralrimiv 3141	. . 3 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))
8	2	isfcls 23907	. . 3 ⊢ (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑥 ∈ 𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
9	1, 3, 7, 8	syl3anbrc 1341	. 2 ⊢ (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹))
10	9	ssriv 3983	1 ⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)

Syntax hints:   ∈ wcel 2099  ∀wral 3057   ⊆ wss 3945  ∪ cuni 4904  ‘cfv 6543  (class class class)co 7415  Topctop 22789  clsccl 22916  Filcfil 23743   fLim cflim 23832   fClus cfcls 23834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-fbas 21270  df-top 22790  df-cld 22917  df-ntr 22918  df-cls 22919  df-nei 22996  df-fil 23744  df-flim 23837  df-fcls 23839

This theorem is referenced by:  fclsfnflim  23925  flimfnfcls  23926  uffclsflim  23929  flfssfcf  23936  cnpfcf  23939  cfilfcls  25196