Theorem isfcls2 24021

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) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

Distinct variable groups:   𝐴,𝑠   𝐹,𝑠   𝐽,𝑠   𝑋,𝑠

Proof of Theorem isfcls2

Step	Hyp	Ref	Expression
1		topontop 22919	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		toponuni 22920	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
3	2	fveq2d 6910	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘∪ 𝐽))
4	3	eleq2d 2827	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐽)))
5	4	biimpa 476	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘∪ 𝐽))
6		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	isfcls 24017	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
8		df-3an 1089	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
9	7, 8	bitri 275	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
10	9	baib 535	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
11	1, 5, 10	syl2an2r 685	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3061  ∪ cuni 4907  ‘cfv 6561  (class class class)co 7431  Topctop 22899  TopOnctopon 22916  clsccl 23026  Filcfil 23853   fClus cfcls 23944

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-topon 22917  df-fil 23854  df-fcls 23949

This theorem is referenced by:  fclsopn  24022  fclsss2  24031