Theorem isfcls2 24127

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 23027	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		toponuni 23028	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
3	2	fveq2d 6875	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘∪ 𝐽))
4	3	eleq2d 2851	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐽)))
5	4	biimpa 481	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘∪ 𝐽))
6		eqid 2765	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	isfcls 24123	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
8		df-3an 1103	. . . 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 278	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
10	9	baib 544	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐽)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
11	1, 5, 10	syl2an2r 697	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2145  ∀wral 3079  ∪ cuni 4867  ‘cfv 6525  (class class class)co 7400  Topctop 23007  TopOnctopon 23024  clsccl 23132  Filcfil 23959   fClus cfcls 24050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iin 4954  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-fbas 21476  df-topon 23025  df-fil 23960  df-fcls 24055

This theorem is referenced by:  fclsopn  24128  fclsss2  24137