Theorem isfcls 23896

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 468	. 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 6891	. . . . . . . 8 ⊢ (Fil‘𝑋) ⊆ ∪ ran Fil
3	2	sseli 3942	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ ∪ ran Fil)
4		filunibas 23768	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
5	4	eqcomd 2735	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐹)
6	3, 5	jca 511	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹))
7		filunirn 23769	. . . . . . 7 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))
8		fveq2 6858	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝐹 → (Fil‘𝑋) = (Fil‘∪ 𝐹))
9	8	eleq2d 2814	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐹)))
10	9	biimparc 479	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
11	7, 10	sylanb 581	. . . . . 6 ⊢ ((𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12	6, 11	impbii 209	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹))
13	12	anbi2i 623	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
14	13	anbi1i 624	. . 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 1088	. . 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 468	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
17	16	anbi1i 624	. . 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 303	. 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 23828	. . . 4 ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
20	19	elmpocl 7630	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
21		fclsval.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
22	21	fclsval 23895	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐹)) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
23	7, 22	sylan2b 594	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
24	23	eleq2d 2814	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ 𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅)))
25		n0i 4303	. . . . . . 7 ⊢ (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) → ¬ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
26		iffalse 4497	. . . . . . 7 ⊢ (¬ 𝑋 = ∪ 𝐹 → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
27	25, 26	nsyl2 141	. . . . . 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 562	. . . 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 4494	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
31	30	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
32	31	eleq2d 2814	. . . . . 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 3468	. . . . . . . 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 23749	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ≠ ∅)
36	7, 35	sylbi 217	. . . . . . . . . 10 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ≠ ∅)
37	36	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ≠ ∅)
38		r19.2z 4458	. . . . . . . . . 10 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
39	38	ex 412	. . . . . . . . 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 3468	. . . . . . . . 9 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
42	41	rexlimivw 3130	. . . . . . . 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 4960	. . . . . . . 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 379	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
47	32, 46	bitrd 279	. . . . 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 579	. . . 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 305	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
50	20, 49	biadanii 821	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
51	1, 18, 50	3bitr4ri 304	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447  ∅c0 4296  ifcif 4488  ∪ cuni 4871  ∩ ciin 4956  ran crn 5639  ‘cfv 6511  (class class class)co 7387  Topctop 22780  clsccl 22905  Filcfil 23732   fClus cfcls 23823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-fbas 21261  df-fil 23733  df-fcls 23828

This theorem is referenced by:  fclsfil  23897  fclstop  23898  isfcls2  23900  fclssscls  23905  flimfcls  23913