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Theorem flimfcls 22023
 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 21962 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)
2 eqid 2752 . . . 4 𝐽 = 𝐽
32flimfil 21966 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐽))
4 flimclsi 21975 . . . . . 6 (𝑥𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑥))
54sseld 3735 . . . . 5 (𝑥𝐹 → (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
65com12 32 . . . 4 (𝑎 ∈ (𝐽 fLim 𝐹) → (𝑥𝐹𝑎 ∈ ((cls‘𝐽)‘𝑥)))
76ralrimiv 3095 . . 3 (𝑎 ∈ (𝐽 fLim 𝐹) → ∀𝑥𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥))
82isfcls 22006 . . 3 (𝑎 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑥𝐹 𝑎 ∈ ((cls‘𝐽)‘𝑥)))
91, 3, 7, 8syl3anbrc 1426 . 2 (𝑎 ∈ (𝐽 fLim 𝐹) → 𝑎 ∈ (𝐽 fClus 𝐹))
109ssriv 3740 1 (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2131  ∀wral 3042   ⊆ wss 3707  ∪ cuni 4580  ‘cfv 6041  (class class class)co 6805  Topctop 20892  clsccl 21016  Filcfil 21842   fLim cflim 21931   fClus cfcls 21933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-fbas 19937  df-top 20893  df-cld 21017  df-ntr 21018  df-cls 21019  df-nei 21096  df-fil 21843  df-flim 21936  df-fcls 21938 This theorem is referenced by:  fclsfnflim  22024  flimfnfcls  22025  uffclsflim  22028  flfssfcf  22035  cnpfcf  22038  cfilfcls  23264
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