MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fclscmpi Structured version   Visualization version   GIF version

Theorem fclscmpi 21955
Description: Forward direction of fclscmp 21956. 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 21321 . . . 4 (𝐽 ∈ Comp → 𝐽 ∈ Top)
2 flimfnfcls.x . . . . . 6 𝑋 = 𝐽
32fclsval 21934 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅))
4 eqid 2724 . . . . . 6 𝑋 = 𝑋
54iftruei 4201 . . . . 5 if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅) = 𝑥𝐹 ((cls‘𝐽)‘𝑥)
63, 5syl6eq 2774 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
71, 6sylan 489 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
8 fvex 6314 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
98dfiin3 5488 . . 3 𝑥𝐹 ((cls‘𝐽)‘𝑥) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))
107, 9syl6eq 2774 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11 simpl 474 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
1211adantr 472 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Comp)
1312, 1syl 17 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Top)
14 filelss 21778 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) → 𝑥𝑋)
1514adantll 752 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝑋)
162clscld 20974 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
1713, 15, 16syl2anc 696 . . . . 5 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
18 eqid 2724 . . . . 5 (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) = (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))
1917, 18fmptd 6500 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
20 frn 6166 . . . 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 479 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
2322adantr 472 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐹 ∈ (Fil‘𝑋))
24 simpr 479 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝐹)
252clsss3 20986 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
2613, 15, 25syl2anc 696 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
272sscls 20983 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
2813, 15, 27syl2anc 696 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
29 filss 21779 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑋𝑥 ⊆ ((cls‘𝐽)‘𝑥))) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
3023, 24, 26, 28, 29syl13anc 1441 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
3130, 18fmptd 6500 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹𝐹)
32 frn 6166 . . . . . . 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 8446 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
3522, 33, 34syl2anc 696 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
36 filfi 21785 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
3722, 36syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘𝐹) = 𝐹)
3835, 37sseqtrd 3747 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
39 0nelfil 21775 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
4022, 39syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
4138, 40ssneldd 3712 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))))
42 cmpfii 21335 . . 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 1439 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
4410, 43eqnetrd 2963 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1596  wcel 2103  wne 2896  wss 3680  c0 4023  ifcif 4194   cuni 4544   cint 4583   ciin 4629  cmpt 4837  ran crn 5219  wf 5997  cfv 6001  (class class class)co 6765  ficfi 8432  Topctop 20821  Clsdccld 20943  clsccl 20945  Compccmp 21312  Filcfil 21771   fClus cfcls 21862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-oadd 7684  df-er 7862  df-map 7976  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fi 8433  df-fbas 19866  df-top 20822  df-cld 20946  df-cls 20948  df-cmp 21313  df-fil 21772  df-fcls 21867
This theorem is referenced by:  fclscmp  21956  ufilcmp  21958  relcmpcmet  23236
  Copyright terms: Public domain W3C validator