Theorem fclsfil 22907

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.

Step	Hyp	Ref	Expression
1		fclsval.x	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	isfcls 22906	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
3	2	simp2bi 1148	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∪ cuni 4819  ‘cfv 6380  (class class class)co 7213  Topctop 21790  clsccl 21915  Filcfil 22742   fClus cfcls 22833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-fbas 20360  df-fil 22743  df-fcls 22838

This theorem is referenced by:  fclstopon  22909  fclsopni  22912  fclselbas  22913  fclsfnflim  22924  cnpfcfi  22937