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Theorem fclscmpi 21643
 Description: Forward direction of fclscmp 21644. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypothesis
Ref Expression
flimfnfcls.x 𝑋 = 𝐽
Assertion
Ref Expression
fclscmpi ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)

Proof of Theorem fclscmpi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cmptop 21008 . . . 4 (𝐽 ∈ Comp → 𝐽 ∈ Top)
2 flimfnfcls.x . . . . . 6 𝑋 = 𝐽
32fclsval 21622 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅))
4 eqid 2610 . . . . . 6 𝑋 = 𝑋
54iftruei 4043 . . . . 5 if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅) = 𝑥𝐹 ((cls‘𝐽)‘𝑥)
63, 5syl6eq 2660 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
71, 6sylan 487 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
8 fvex 6113 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
98dfiin3 5302 . . 3 𝑥𝐹 ((cls‘𝐽)‘𝑥) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))
107, 9syl6eq 2660 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11 simpl 472 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
1211adantr 480 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Comp)
1312, 1syl 17 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Top)
14 filelss 21466 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) → 𝑥𝑋)
1514adantll 746 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝑋)
162clscld 20661 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
1713, 15, 16syl2anc 691 . . . . 5 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
18 eqid 2610 . . . . 5 (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) = (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))
1917, 18fmptd 6292 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
20 frn 5966 . . . 4 ((𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
2119, 20syl 17 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
22 simpr 476 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
2322adantr 480 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐹 ∈ (Fil‘𝑋))
24 simpr 476 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝐹)
252clsss3 20673 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
2613, 15, 25syl2anc 691 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
272sscls 20670 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
2813, 15, 27syl2anc 691 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
29 filss 21467 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑋𝑥 ⊆ ((cls‘𝐽)‘𝑥))) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
3023, 24, 26, 28, 29syl13anc 1320 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
3130, 18fmptd 6292 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹𝐹)
32 frn 5966 . . . . . . 7 ((𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹𝐹 → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
3331, 32syl 17 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
34 fiss 8213 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
3522, 33, 34syl2anc 691 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
36 filfi 21473 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
3722, 36syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘𝐹) = 𝐹)
3835, 37sseqtrd 3604 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
39 0nelfil 21463 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
4022, 39syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
4138, 40ssneldd 3571 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))))
42 cmpfii 21022 . . 3 ((𝐽 ∈ Comp ∧ ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)))) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
4311, 21, 41, 42syl3anc 1318 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
4410, 43eqnetrd 2849 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ wss 3540  ∅c0 3874  ifcif 4036  ∪ cuni 4372  ∩ cint 4410  ∩ ciin 4456   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ficfi 8199  Topctop 20517  Clsdccld 20630  clsccl 20632  Compccmp 20999  Filcfil 21459   fClus cfcls 21550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-fbas 19564  df-top 20521  df-cld 20633  df-cls 20635  df-cmp 21000  df-fil 21460  df-fcls 21555 This theorem is referenced by:  fclscmp  21644  ufilcmp  21646  relcmpcmet  22923
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