MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fclstop Structured version   Visualization version   GIF version

Theorem fclstop 23735
Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclstop (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)

Proof of Theorem fclstop
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 eqid 2730 . . 3 𝐽 = 𝐽
21isfcls 23733 . 2 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
32simp1bi 1143 1 (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104  wral 3059   cuni 4907  cfv 6542  (class class class)co 7411  Topctop 22615  clsccl 22742  Filcfil 23569   fClus cfcls 23660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-fil 23570  df-fcls 23665
This theorem is referenced by:  fclstopon  23736  fclsneii  23741  fclsfnflim  23751  flimfnfcls  23752
  Copyright terms: Public domain W3C validator