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Theorem fclsval 22617
 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, 𝑡𝐹 ((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.
StepHypRef Expression
1 simpl 486 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
2 fvssunirn 6678 . . . . 5 (Fil‘𝑌) ⊆ ran Fil
32sseli 3914 . . . 4 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ran Fil)
43adantl 485 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ran Fil)
5 filn0 22471 . . . . . 6 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ≠ ∅)
65adantl 485 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ≠ ∅)
7 fvex 6662 . . . . . 6 ((cls‘𝐽)‘𝑡) ∈ V
87rgenw 3121 . . . . 5 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V
9 iinexg 5211 . . . . 5 ((𝐹 ≠ ∅ ∧ ∀𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
106, 8, 9sylancl 589 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
11 0ex 5178 . . . 4 ∅ ∈ V
12 ifcl 4472 . . . 4 (( 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V ∧ ∅ ∈ V) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
1310, 11, 12sylancl 589 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
14 unieq 4814 . . . . . . 7 (𝑗 = 𝐽 𝑗 = 𝐽)
15 fclsval.x . . . . . . 7 𝑋 = 𝐽
1614, 15eqtr4di 2854 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝑋)
17 unieq 4814 . . . . . 6 (𝑓 = 𝐹 𝑓 = 𝐹)
1816, 17eqeqan12d 2818 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → ( 𝑗 = 𝑓𝑋 = 𝐹))
19 iineq1 4901 . . . . . . 7 (𝑓 = 𝐹 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
2019adantl 485 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
21 simpll 766 . . . . . . . . 9 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → 𝑗 = 𝐽)
2221fveq2d 6653 . . . . . . . 8 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → (cls‘𝑗) = (cls‘𝐽))
2322fveq1d 6651 . . . . . . 7 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → ((cls‘𝑗)‘𝑡) = ((cls‘𝐽)‘𝑡))
2423iineq2dv 4909 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝐹 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2520, 24eqtrd 2836 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2618, 25ifbieq1d 4451 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
27 df-fcls 22550 . . . 4 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅))
2826, 27ovmpoga 7287 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
291, 4, 13, 28syl3anc 1368 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
30 filunibas 22490 . . . . 5 (𝐹 ∈ (Fil‘𝑌) → 𝐹 = 𝑌)
3130eqeq2d 2812 . . . 4 (𝐹 ∈ (Fil‘𝑌) → (𝑋 = 𝐹𝑋 = 𝑌))
3231adantl 485 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝑋 = 𝐹𝑋 = 𝑌))
3332ifbid 4450 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) = if(𝑋 = 𝑌, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
3429, 33eqtrd 2836 1 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑌, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  Vcvv 3444  ∅c0 4246  ifcif 4428  ∪ cuni 4803  ∩ ciin 4885  ran crn 5524  ‘cfv 6328  (class class class)co 7139  Topctop 21502  clsccl 21627  Filcfil 22454   fClus cfcls 22545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-fbas 20092  df-fil 22455  df-fcls 22550 This theorem is referenced by:  isfcls  22618  fclscmpi  22638
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