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Theorem fclsval 21793
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 473 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
2 fvssunirn 6204 . . . . 5 (Fil‘𝑌) ⊆ ran Fil
32sseli 3591 . . . 4 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ran Fil)
43adantl 482 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ran Fil)
5 filn0 21647 . . . . . 6 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ≠ ∅)
65adantl 482 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ≠ ∅)
7 fvex 6188 . . . . . 6 ((cls‘𝐽)‘𝑡) ∈ V
87rgenw 2921 . . . . 5 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V
9 iinexg 4815 . . . . 5 ((𝐹 ≠ ∅ ∧ ∀𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
106, 8, 9sylancl 693 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
11 0ex 4781 . . . 4 ∅ ∈ V
12 ifcl 4121 . . . 4 (( 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V ∧ ∅ ∈ V) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
1310, 11, 12sylancl 693 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
14 unieq 4435 . . . . . . 7 (𝑗 = 𝐽 𝑗 = 𝐽)
15 fclsval.x . . . . . . 7 𝑋 = 𝐽
1614, 15syl6eqr 2672 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝑋)
17 unieq 4435 . . . . . 6 (𝑓 = 𝐹 𝑓 = 𝐹)
1816, 17eqeqan12d 2636 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → ( 𝑗 = 𝑓𝑋 = 𝐹))
19 iineq1 4526 . . . . . . 7 (𝑓 = 𝐹 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
2019adantl 482 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
21 simpll 789 . . . . . . . . 9 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → 𝑗 = 𝐽)
2221fveq2d 6182 . . . . . . . 8 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → (cls‘𝑗) = (cls‘𝐽))
2322fveq1d 6180 . . . . . . 7 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → ((cls‘𝑗)‘𝑡) = ((cls‘𝐽)‘𝑡))
2423iineq2dv 4534 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝐹 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2520, 24eqtrd 2654 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2618, 25ifbieq1d 4100 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
27 df-fcls 21726 . . . 4 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅))
2826, 27ovmpt2ga 6775 . . 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 1324 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
30 filunibas 21666 . . . . 5 (𝐹 ∈ (Fil‘𝑌) → 𝐹 = 𝑌)
3130eqeq2d 2630 . . . 4 (𝐹 ∈ (Fil‘𝑌) → (𝑋 = 𝐹𝑋 = 𝑌))
3231adantl 482 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝑋 = 𝐹𝑋 = 𝑌))
3332ifbid 4099 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) = if(𝑋 = 𝑌, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
3429, 33eqtrd 2654 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 196  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  Vcvv 3195  c0 3907  ifcif 4077   cuni 4427   ciin 4512  ran crn 5105  cfv 5876  (class class class)co 6635  Topctop 20679  clsccl 20803  Filcfil 21630   fClus cfcls 21721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-fbas 19724  df-fil 21631  df-fcls 21726
This theorem is referenced by:  isfcls  21794  fclscmpi  21814
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