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Theorem isfcls2 23446
Description: A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
isfcls2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
Distinct variable groups:   𝐴,𝑠   𝐹,𝑠   𝐽,𝑠   𝑋,𝑠

Proof of Theorem isfcls2
StepHypRef Expression
1 topontop 22344 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 toponuni 22345 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
32fveq2d 6882 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘ 𝐽))
43eleq2d 2818 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘ 𝐽)))
54biimpa 477 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘ 𝐽))
6 eqid 2731 . . . . 5 𝐽 = 𝐽
76isfcls 23442 . . . 4 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
8 df-3an 1089 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽)) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
97, 8bitri 274 . . 3 (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽)) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
109baib 536 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘ 𝐽)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
111, 5, 10syl2an2r 683 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wcel 2106  wral 3060   cuni 4901  cfv 6532  (class class class)co 7393  Topctop 22324  TopOnctopon 22341  clsccl 22451  Filcfil 23278   fClus cfcls 23369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-fbas 20875  df-topon 22342  df-fil 23279  df-fcls 23374
This theorem is referenced by:  fclsopn  23447  fclsss2  23456
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