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Theorem flimfcls 23950
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 23889 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2728 . . . 4 𝐽 = 𝐽
32flimfil 23893 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimclsi 23902 . . . . . 6 (𝑥𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥))
54sseld 3981 . . . . 5 (𝑥𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
65com12 32 . . . 4 (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥𝐹𝑎 ∈ ((cls‘𝐽)‘𝑥)))
76ralrimiv 3142 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))
82isfcls 23933 . . 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 3986 1 (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  wral 3058  wss 3949   cuni 4912  cfv 6553  (class class class)co 7426  Topctop 22815  clsccl 22942  Filcfil 23769   fLim cflim 23858   fClus cfcls 23860
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-fbas 21283  df-top 22816  df-cld 22943  df-ntr 22944  df-cls 22945  df-nei 23022  df-fil 23770  df-flim 23863  df-fcls 23865
This theorem is referenced by:  fclsfnflim  23951  flimfnfcls  23952  uffclsflim  23955  flfssfcf  23962  cnpfcf  23965  cfilfcls  25222
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