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Theorem fclsfil 22034
Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypothesis
Ref Expression
fclsval.x 𝑋 = 𝐽
Assertion
Ref Expression
fclsfil (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Proof of Theorem fclsfil
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 fclsval.x . . 3 𝑋 = 𝐽
21isfcls 22033 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
32simp2bi 1140 1 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wral 3061   cuni 4574  cfv 6031  (class class class)co 6793  Topctop 20918  clsccl 21043  Filcfil 21869   fClus cfcls 21960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-fbas 19958  df-fil 21870  df-fcls 21965
This theorem is referenced by:  fclstopon  22036  fclsopni  22039  fclselbas  22040  fclsfnflim  22051  cnpfcfi  22064
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