Theorem fclstopon 23999

Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
fclstopon	⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

Proof of Theorem fclstopon

Step	Hyp	Ref	Expression
1		fclstop 23998	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
2		istopon 22899	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
3	2	baib 541	. . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = ∪ 𝐽))
4	1, 3	syl 17	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = ∪ 𝐽))
5		eqid 2741	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	fclsfil 23997	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
7		fveq2 6831	. . . . 5 ⊢ (𝑋 = ∪ 𝐽 → (Fil‘𝑋) = (Fil‘∪ 𝐽))
8	7	eleq2d 2827	. . . 4 ⊢ (𝑋 = ∪ 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐽)))
9	6, 8	syl5ibrcom 249	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 → 𝐹 ∈ (Fil‘𝑋)))
10		filunibas 23868	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽)
11	6, 10	syl 17	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∪ 𝐹 = ∪ 𝐽)
12		filunibas 23868	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
13	12	eqeq1d 2743	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∪ 𝐹 = ∪ 𝐽 ↔ 𝑋 = ∪ 𝐽))
14	11, 13	syl5ibcom 247	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐽))
15	9, 14	impbid 214	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 ↔ 𝐹 ∈ (Fil‘𝑋)))
16	4, 15	bitrd 281	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548   ∈ wcel 2121  ∪ cuni 4841  ‘cfv 6489  (class class class)co 7360  Topctop 22880  TopOnctopon 22897  Filcfil 23832   fClus cfcls 23923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-fbas 21348  df-topon 22898  df-fil 23833  df-fcls 23928

This theorem is referenced by:  fclsopni  24002  fclselbas  24003  fclsss1  24009  fclsss2  24010  fclscf  24012