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Theorem flimfcls 23885
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 23824 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2726 . . . 4 𝐽 = 𝐽
32flimfil 23828 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimclsi 23837 . . . . . 6 (𝑥𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥))
54sseld 3976 . . . . 5 (𝑥𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
65com12 32 . . . 4 (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥𝐹𝑎 ∈ ((cls‘𝐽)‘𝑥)))
76ralrimiv 3139 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))
82isfcls 23868 . . 3 (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑥𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
91, 3, 7, 8syl3anbrc 1340 . 2 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹))
109ssriv 3981 1 (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  wral 3055  wss 3943   cuni 4902  cfv 6537  (class class class)co 7405  Topctop 22750  clsccl 22877  Filcfil 23704   fLim cflim 23793   fClus cfcls 23795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-fbas 21237  df-top 22751  df-cld 22878  df-ntr 22879  df-cls 22880  df-nei 22957  df-fil 23705  df-flim 23798  df-fcls 23800
This theorem is referenced by:  fclsfnflim  23886  flimfnfcls  23887  uffclsflim  23890  flfssfcf  23897  cnpfcf  23900  cfilfcls  25157
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