Theorem fclstop 22303

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.

Step	Hyp	Ref	Expression
1		eqid 2795	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isfcls 22301	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
3	2	simp1bi 1138	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2081  ∀wral 3105  ∪ cuni 4745  ‘cfv 6225  (class class class)co 7016  Topctop 21185  clsccl 21310  Filcfil 22137   fClus cfcls 22228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-int 4783  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-fbas 20224  df-fil 22138  df-fcls 22233

This theorem is referenced by:  fclstopon  22304  fclsneii  22309  fclsfnflim  22319  flimfnfcls  22320