Theorem isfcls2 23988

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 22888	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		toponuni 22889	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
3	2	fveq2d 6838	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘∪ 𝐽))
4	3	eleq2d 2823	. . 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 23984	. . . 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 686	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 2114  ∀wral 3052  ∪ cuni 4851  ‘cfv 6492  (class class class)co 7360  Topctop 22868  TopOnctopon 22885  clsccl 22993  Filcfil 23820   fClus cfcls 23911

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-fbas 21341  df-topon 22886  df-fil 23821  df-fcls 23916

This theorem is referenced by:  fclsopn  23989  fclsss2  23998