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Theorem fclsval 22021
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 470 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
2 fvssunirn 6434 . . . . 5 (Fil‘𝑌) ⊆ ran Fil
32sseli 3791 . . . 4 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ran Fil)
43adantl 469 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ran Fil)
5 filn0 21875 . . . . . 6 (𝐹 ∈ (Fil‘𝑌) → 𝐹 ≠ ∅)
65adantl 469 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ≠ ∅)
7 fvex 6418 . . . . . 6 ((cls‘𝐽)‘𝑡) ∈ V
87rgenw 3111 . . . . 5 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V
9 iinexg 5013 . . . . 5 ((𝐹 ≠ ∅ ∧ ∀𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
106, 8, 9sylancl 576 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
11 0ex 4981 . . . 4 ∅ ∈ V
12 ifcl 4320 . . . 4 (( 𝑡𝐹 ((cls‘𝐽)‘𝑡) ∈ V ∧ ∅ ∈ V) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
1310, 11, 12sylancl 576 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
14 unieq 4634 . . . . . . 7 (𝑗 = 𝐽 𝑗 = 𝐽)
15 fclsval.x . . . . . . 7 𝑋 = 𝐽
1614, 15syl6eqr 2857 . . . . . 6 (𝑗 = 𝐽 𝑗 = 𝑋)
17 unieq 4634 . . . . . 6 (𝑓 = 𝐹 𝑓 = 𝐹)
1816, 17eqeqan12d 2821 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → ( 𝑗 = 𝑓𝑋 = 𝐹))
19 iineq1 4723 . . . . . . 7 (𝑓 = 𝐹 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
2019adantl 469 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝑗)‘𝑡))
21 simpll 774 . . . . . . . . 9 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → 𝑗 = 𝐽)
2221fveq2d 6409 . . . . . . . 8 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → (cls‘𝑗) = (cls‘𝐽))
2322fveq1d 6407 . . . . . . 7 (((𝑗 = 𝐽𝑓 = 𝐹) ∧ 𝑡𝐹) → ((cls‘𝑗)‘𝑡) = ((cls‘𝐽)‘𝑡))
2423iineq2dv 4731 . . . . . 6 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝐹 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2520, 24eqtrd 2839 . . . . 5 ((𝑗 = 𝐽𝑓 = 𝐹) → 𝑡𝑓 ((cls‘𝑗)‘𝑡) = 𝑡𝐹 ((cls‘𝐽)‘𝑡))
2618, 25ifbieq1d 4299 . . . 4 ((𝑗 = 𝐽𝑓 = 𝐹) → if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
27 df-fcls 21954 . . . 4 fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑡𝑓 ((cls‘𝑗)‘𝑡), ∅))
2826, 27ovmpt2ga 7017 . . 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 1483 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
30 filunibas 21894 . . . . 5 (𝐹 ∈ (Fil‘𝑌) → 𝐹 = 𝑌)
3130eqeq2d 2815 . . . 4 (𝐹 ∈ (Fil‘𝑌) → (𝑋 = 𝐹𝑋 = 𝑌))
3231adantl 469 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝑋 = 𝐹𝑋 = 𝑌))
3332ifbid 4298 . 2 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = 𝐹, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅) = if(𝑋 = 𝑌, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))
3429, 33eqtrd 2839 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 197  wa 384   = wceq 1637  wcel 2158  wne 2977  wral 3095  Vcvv 3390  c0 4113  ifcif 4276   cuni 4626   ciin 4709  ran crn 5309  cfv 6098  (class class class)co 6871  Topctop 20907  clsccl 21032  Filcfil 21858   fClus cfcls 21949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-int 4666  df-iin 4711  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fv 6106  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-fbas 19947  df-fil 21859  df-fcls 21954
This theorem is referenced by:  isfcls  22022  fclscmpi  22042
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