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Theorem fclsval 23932
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 481 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
2 fvssunirn 6935 . . . . 5 (Fil‘𝑌) ⊆ ran Fil
32sseli 3978 . . . 4 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ran Fil)
43adantl 480 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ran Fil)
5 filn0 23786 . . . . . 6 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ≠ ∅)
65adantl 480 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ≠ ∅)
7 fvex 6915 . . . . . 6 ((cls‘𝐽)‘𝑡) ∈ V
87rgenw 3062 . . . . 5 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V
9 iinexg 5347 . . . . 5 ((𝐹 ≠ ∅ ∧ ∀𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
106, 8, 9sylancl 584 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
11 0ex 5311 . . . 4 ∅ ∈ V
12 ifcl 4577 . . . 4 (( 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V ∧ ∅ ∈ V) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
1310, 11, 12sylancl 584 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
14 unieq 4923 . . . . . . 7 (𝑗 = 𝐽 𝑗 = 𝐽)
15 fclsval.x . . . . . . 7 𝑋 = 𝐽
1614, 15eqtr4di 2786 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝑋)
17 unieq 4923 . . . . . 6 (𝑓 = 𝐹 𝑓 = 𝐹)
1816, 17eqeqan12d 2742 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → ( 𝑗 = 𝑓𝑋 = 𝐹))
19 iineq1 5017 . . . . . . 7 (𝑓 = 𝐹 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
2019adantl 480 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
21 simpll 765 . . . . . . . . 9 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → 𝑗 = 𝐽)
2221fveq2d 6906 . . . . . . . 8 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → (cls‘𝑗) = (cls‘𝐽))
2322fveq1d 6904 . . . . . . 7 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → ((cls‘𝑗)‘𝑡) = ((cls‘𝐽)‘𝑡))
2423iineq2dv 5025 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝐹 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2520, 24eqtrd 2768 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2618, 25ifbieq1d 4556 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
27 df-fcls 23865 . . . 4 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅))
2826, 27ovmpoga 7581 . . 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 23805 . . . . 5 (𝐹 ∈ (Fil‘𝑌) → 𝐹 = 𝑌)
3130eqeq2d 2739 . . . 4 (𝐹 ∈ (Fil‘𝑌) → (𝑋 = 𝐹𝑋 = 𝑌))
3231adantl 480 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝑋 = 𝐹𝑋 = 𝑌))
3332ifbid 4555 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) = if(𝑋 = 𝑌, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
3429, 33eqtrd 2768 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 205  wa 394   = wceq 1533  wcel 2098  wne 2937  wral 3058  Vcvv 3473  c0 4326  ifcif 4532   cuni 4912   ciin 5001  ran crn 5683  cfv 6553  (class class class)co 7426  Topctop 22815  clsccl 22942  Filcfil 23769   fClus cfcls 23860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-fbas 21283  df-fil 23770  df-fcls 23865
This theorem is referenced by:  isfcls  23933  fclscmpi  23953
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