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Theorem fclsval 22032
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 468 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
2 fvssunirn 6358 . . . . 5 (Fil‘𝑌) ⊆ ran Fil
32sseli 3748 . . . 4 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ran Fil)
43adantl 467 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ran Fil)
5 filn0 21886 . . . . . 6 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ≠ ∅)
65adantl 467 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ≠ ∅)
7 fvex 6342 . . . . . 6 ((cls‘𝐽)‘𝑡) ∈ V
87rgenw 3073 . . . . 5 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V
9 iinexg 4955 . . . . 5 ((𝐹 ≠ ∅ ∧ ∀𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
106, 8, 9sylancl 574 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
11 0ex 4924 . . . 4 ∅ ∈ V
12 ifcl 4269 . . . 4 (( 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V ∧ ∅ ∈ V) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
1310, 11, 12sylancl 574 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
14 unieq 4582 . . . . . . 7 (𝑗 = 𝐽 𝑗 = 𝐽)
15 fclsval.x . . . . . . 7 𝑋 = 𝐽
1614, 15syl6eqr 2823 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝑋)
17 unieq 4582 . . . . . 6 (𝑓 = 𝐹 𝑓 = 𝐹)
1816, 17eqeqan12d 2787 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → ( 𝑗 = 𝑓𝑋 = 𝐹))
19 iineq1 4669 . . . . . . 7 (𝑓 = 𝐹 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
2019adantl 467 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
21 simpll 750 . . . . . . . . 9 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → 𝑗 = 𝐽)
2221fveq2d 6336 . . . . . . . 8 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → (cls‘𝑗) = (cls‘𝐽))
2322fveq1d 6334 . . . . . . 7 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → ((cls‘𝑗)‘𝑡) = ((cls‘𝐽)‘𝑡))
2423iineq2dv 4677 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝐹 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2520, 24eqtrd 2805 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2618, 25ifbieq1d 4248 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
27 df-fcls 21965 . . . 4 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅))
2826, 27ovmpt2ga 6937 . . 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 1476 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
30 filunibas 21905 . . . . 5 (𝐹 ∈ (Fil‘𝑌) → 𝐹 = 𝑌)
3130eqeq2d 2781 . . . 4 (𝐹 ∈ (Fil‘𝑌) → (𝑋 = 𝐹𝑋 = 𝑌))
3231adantl 467 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝑋 = 𝐹𝑋 = 𝑌))
3332ifbid 4247 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) = if(𝑋 = 𝑌, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
3429, 33eqtrd 2805 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 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  Vcvv 3351  c0 4063  ifcif 4225   cuni 4574   ciin 4655  ran crn 5250  cfv 6031  (class class class)co 6793  Topctop 20918  clsccl 21043  Filcfil 21869   fClus cfcls 21960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-fbas 19958  df-fil 21870  df-fcls 21965
This theorem is referenced by:  isfcls  22033  fclscmpi  22053
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