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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.
StepHypRef Expression
1 flimtop 23863 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2728 . . . 4 𝐽 = 𝐽
32flimfil 23867 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimclsi 23876 . . . . . 6 (𝑥𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥))
54sseld 3978 . . . . 5 (𝑥𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
65com12 32 . . . 4 (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥𝐹𝑎 ∈ ((cls‘𝐽)‘𝑥)))
76ralrimiv 3141 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))
82isfcls 23907 . . 3 (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑥𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
91, 3, 7, 8syl3anbrc 1341 . 2 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹))
109ssriv 3983 1 (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
Colors of variables: wff setvar class
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
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