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Theorem fclscmpi 22050
Description: Forward direction of fclscmp 22051. 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 21416 . . . 4 (𝐽 ∈ Comp → 𝐽 ∈ Top)
2 flimfnfcls.x . . . . . 6 𝑋 = 𝐽
32fclsval 22029 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅))
4 eqid 2813 . . . . . 6 𝑋 = 𝑋
54iftruei 4293 . . . . 5 if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅) = 𝑥𝐹 ((cls‘𝐽)‘𝑥)
63, 5syl6eq 2863 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
71, 6sylan 571 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
8 fvex 6424 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
98dfiin3 5589 . . 3 𝑥𝐹 ((cls‘𝐽)‘𝑥) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))
107, 9syl6eq 2863 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11 simpl 470 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
1211adantr 468 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Comp)
1312, 1syl 17 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Top)
14 filelss 21873 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) → 𝑥𝑋)
1514adantll 696 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝑋)
162clscld 21069 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
1713, 15, 16syl2anc 575 . . . . 5 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
1817fmpttd 6610 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
1918frnd 6266 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
20 simpr 473 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
2120adantr 468 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐹 ∈ (Fil‘𝑋))
22 simpr 473 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝐹)
232clsss3 21081 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
2413, 15, 23syl2anc 575 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
252sscls 21078 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
2613, 15, 25syl2anc 575 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
27 filss 21874 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑋𝑥 ⊆ ((cls‘𝐽)‘𝑥))) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
2821, 22, 24, 26, 27syl13anc 1484 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
2928fmpttd 6610 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹𝐹)
3029frnd 6266 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
31 fiss 8572 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
3220, 30, 31syl2anc 575 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
33 filfi 21880 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
3420, 33syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘𝐹) = 𝐹)
3532, 34sseqtrd 3845 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
36 0nelfil 21870 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
3720, 36syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
3835, 37ssneldd 3808 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))))
39 cmpfii 21430 . . 3 ((𝐽 ∈ Comp ∧ ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)))) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
4011, 19, 38, 39syl3anc 1483 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
4110, 40eqnetrd 3052 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1637  wcel 2157  wne 2985  wss 3776  c0 4123  ifcif 4286   cuni 4637   cint 4676   ciin 4720  cmpt 4930  ran crn 5319  cfv 6104  (class class class)co 6877  ficfi 8558  Topctop 20915  Clsdccld 21038  clsccl 21040  Compccmp 21407  Filcfil 21866   fClus cfcls 21957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-iin 4722  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fi 8559  df-fbas 19954  df-top 20916  df-cld 21041  df-cls 21043  df-cmp 21408  df-fil 21867  df-fcls 21962
This theorem is referenced by:  fclscmp  22051  ufilcmp  22053  relcmpcmet  23332
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