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Theorem flimfcls 22562
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 22501 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2818 . . . 4 𝐽 = 𝐽
32flimfil 22505 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimclsi 22514 . . . . . 6 (𝑥𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥))
54sseld 3963 . . . . 5 (𝑥𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
65com12 32 . . . 4 (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥𝐹𝑎 ∈ ((cls‘𝐽)‘𝑥)))
76ralrimiv 3178 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))
82isfcls 22545 . . 3 (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑥𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
91, 3, 7, 8syl3anbrc 1335 . 2 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹))
109ssriv 3968 1 (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  wral 3135  wss 3933   cuni 4830  cfv 6348  (class class class)co 7145  Topctop 21429  clsccl 21554  Filcfil 22381   fLim cflim 22470   fClus cfcls 22472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-fbas 20470  df-top 21430  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-fil 22382  df-flim 22475  df-fcls 22477
This theorem is referenced by:  fclsfnflim  22563  flimfnfcls  22564  uffclsflim  22567  flfssfcf  22574  cnpfcf  22577  cfilfcls  23804
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