Theorem fclstopon 24036

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 24035	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)
2		istopon 22934	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽))
3	2	baib 535	. . 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 2735	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
6	5	fclsfil 24034	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽))
7		fveq2 6907	. . . . 5 ⊢ (𝑋 = ∪ 𝐽 → (Fil‘𝑋) = (Fil‘∪ 𝐽))
8	7	eleq2d 2825	. . . 4 ⊢ (𝑋 = ∪ 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐽)))
9	6, 8	syl5ibrcom 247	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 → 𝐹 ∈ (Fil‘𝑋)))
10		filunibas 23905	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽)
11	6, 10	syl 17	. . . 4 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → ∪ 𝐹 = ∪ 𝐽)
12		filunibas 23905	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
13	12	eqeq1d 2737	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∪ 𝐹 = ∪ 𝐽 ↔ 𝑋 = ∪ 𝐽))
14	11, 13	syl5ibcom 245	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐽))
15	9, 14	impbid 212	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝑋 = ∪ 𝐽 ↔ 𝐹 ∈ (Fil‘𝑋)))
16	4, 15	bitrd 279	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  ∪ cuni 4912  ‘cfv 6563  (class class class)co 7431  Topctop 22915  TopOnctopon 22932  Filcfil 23869   fClus cfcls 23960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-topon 22933  df-fil 23870  df-fcls 23965

This theorem is referenced by:  fclsopni  24039  fclselbas  24040  fclsss1  24046  fclsss2  24047  fclscf  24049