Theorem isfcls 22324

Description: A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypothesis

Ref	Expression
fclsval.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
isfcls	⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

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

Proof of Theorem isfcls

Dummy variables 𝑓 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		anass 461	. 2 ⊢ ((((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
2		fvssunirn 6530	. . . . . . . 8 ⊢ (Fil‘𝑋) ⊆ ∪ ran Fil
3	2	sseli 3856	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ ∪ ran Fil)
4		filunibas 22196	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
5	4	eqcomd 2784	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐹)
6	3, 5	jca 504	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹))
7		filunirn 22197	. . . . . . 7 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))
8		fveq2 6501	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝐹 → (Fil‘𝑋) = (Fil‘∪ 𝐹))
9	8	eleq2d 2851	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐹)))
10	9	biimparc 472	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
11	7, 10	sylanb 573	. . . . . 6 ⊢ ((𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12	6, 11	impbii 201	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹))
13	12	anbi2i 613	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
14	13	anbi1i 614	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
15		df-3an 1070	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
16		anass 461	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
17	16	anbi1i 614	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
18	14, 15, 17	3bitr4i 295	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
19		df-fcls 22256	. . . 4 ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
20	19	elmpocl 7208	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
21		fclsval.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
22	21	fclsval 22323	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐹)) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
23	7, 22	sylan2b 584	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
24	23	eleq2d 2851	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ 𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅)))
25		n0i 4187	. . . . . . 7 ⊢ (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) → ¬ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
26		iffalse 4360	. . . . . . 7 ⊢ (¬ 𝑋 = ∪ 𝐹 → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
27	25, 26	nsyl2 145	. . . . . 6 ⊢ (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) → 𝑋 = ∪ 𝐹)
28	27	a1i 11	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) → 𝑋 = ∪ 𝐹))
29	28	pm4.71rd 555	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ (𝑋 = ∪ 𝐹 ∧ 𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))))
30		iftrue 4357	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
31	30	adantl 474	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
32	31	eleq2d 2851	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ 𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠)))
33		elex 3433	. . . . . . . 8 ⊢ (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
34	33	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V))
35		filn0 22177	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ≠ ∅)
36	7, 35	sylbi 209	. . . . . . . . . 10 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ≠ ∅)
37	36	ad2antlr 714	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ≠ ∅)
38		r19.2z 4324	. . . . . . . . . 10 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
39	38	ex 405	. . . . . . . . 9 ⊢ (𝐹 ≠ ∅ → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
40	37, 39	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
41		elex 3433	. . . . . . . . 9 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
42	41	rexlimivw 3227	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
43	40, 42	syl6 35	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V))
44		eliin 4798	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
45	44	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ V → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
46	34, 43, 45	pm5.21ndd 372	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
47	32, 46	bitrd 271	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
48	47	pm5.32da 571	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → ((𝑋 = ∪ 𝐹 ∧ 𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅)) ↔ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
49	24, 29, 48	3bitrd 297	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
50	20, 49	biadanii 810	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
51	1, 18, 50	3bitr4ri 296	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2967  ∀wral 3088  ∃wrex 3089  Vcvv 3415  ∅c0 4180  ifcif 4351  ∪ cuni 4713  ∩ ciin 4794  ran crn 5409  ‘cfv 6190  (class class class)co 6978  Topctop 21208  clsccl 21333  Filcfil 22160   fClus cfcls 22251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-int 4751  df-iin 4796  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-fbas 20247  df-fil 22161  df-fcls 22256

This theorem is referenced by:  fclsfil  22325  fclstop  22326  isfcls2  22328  fclssscls  22333  flimfcls  22341