Theorem fclsval 24144

Description: The set of all cluster points 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
fclsval	⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑌, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))

Distinct variable groups:   𝑡,𝐹   𝑡,𝐽

Allowed substitution hints:   𝑋(𝑡)   𝑌(𝑡)

Proof of Theorem fclsval

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

Step	Hyp	Ref	Expression
1		simpl 487	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
2		fvssunirn 6912	. . . . 5 ⊢ (Fil‘𝑌) ⊆ ∪ ran Fil
3	2	sseli 3932	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑌) → 𝐹 ∈ ∪ ran Fil)
4	3	adantl 486	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ∈ ∪ ran Fil)
5		filn0 23998	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑌) → 𝐹 ≠ ∅)
6	5	adantl 486	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ≠ ∅)
7		fvex 6894	. . . . . 6 ⊢ ((cls‘𝐽)‘𝑡) ∈ V
8	7	rgenw 3081	. . . . 5 ⊢ ∀𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V
9		iinexg 5318	. . . . 5 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V) → ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
10	6, 8, 9	sylancl 597	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
11		0ex 5269	. . . 4 ⊢ ∅ ∈ V
12		ifcl 4532	. . . 4 ⊢ ((∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V ∧ ∅ ∈ V) → if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
13	10, 11, 12	sylancl 597	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
14		unieq 4882	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
15		fclsval.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
16	14, 15	eqtr4di 2814	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
17		unieq 4882	. . . . . 6 ⊢ (𝑓 = 𝐹 → ∪ 𝑓 = ∪ 𝐹)
18	16, 17	eqeqan12d 2775	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (∪ 𝑗 = ∪ 𝑓 ↔ 𝑋 = ∪ 𝐹))
19		iineq1 4973	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝑗)‘𝑡))
20	19	adantl 486	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝑗)‘𝑡))
21		simpll 778	. . . . . . . . 9 ⊢ (((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) ∧ 𝑡 ∈ 𝐹) → 𝑗 = 𝐽)
22	21	fveq2d 6885	. . . . . . . 8 ⊢ (((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) ∧ 𝑡 ∈ 𝐹) → (cls‘𝑗) = (cls‘𝐽))
23	22	fveq1d 6883	. . . . . . 7 ⊢ (((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) ∧ 𝑡 ∈ 𝐹) → ((cls‘𝑗)‘𝑡) = ((cls‘𝐽)‘𝑡))
24	23	iineq2dv 4981	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∩ 𝑡 ∈ 𝐹 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡))
25	20, 24	eqtrd 2796	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡))
26	18, 25	ifbieq1d 4511	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡), ∅) = if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
27		df-fcls 24077	. . . 4 ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡), ∅))
28	26, 27	ovmpoga 7564	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
29	1, 4, 13, 28	syl3anc 1396	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
30		filunibas 24017	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑌) → ∪ 𝐹 = 𝑌)
31	30	eqeq2d 2772	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑌) → (𝑋 = ∪ 𝐹 ↔ 𝑋 = 𝑌))
32	31	adantl 486	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝑋 = ∪ 𝐹 ↔ 𝑋 = 𝑌))
33	32	ifbid 4510	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) = if(𝑋 = 𝑌, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
34	29, 33	eqtrd 2796	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑌, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  Vcvv 3453  ∅c0 4285  ifcif 4486  ∪ cuni 4871  ∩ ciin 4956  ran crn 5662  ‘cfv 6536  (class class class)co 7410  Topctop 23029  clsccl 23154  Filcfil 23981   fClus cfcls 24072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-fbas 21498  df-fil 23982  df-fcls 24077

This theorem is referenced by:  isfcls  24145  fclscmpi  24165