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Theorem fclscmpi 22638
Description: Forward direction of fclscmp 22639. 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 22004 . . . 4 (𝐽 ∈ Comp → 𝐽 ∈ Top)
2 flimfnfcls.x . . . . . 6 𝑋 = 𝐽
32fclsval 22617 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅))
4 eqid 2801 . . . . . 6 𝑋 = 𝑋
54iftruei 4435 . . . . 5 if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅) = 𝑥𝐹 ((cls‘𝐽)‘𝑥)
63, 5eqtrdi 2852 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
71, 6sylan 583 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
8 fvex 6662 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
98dfiin3 5807 . . 3 𝑥𝐹 ((cls‘𝐽)‘𝑥) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))
107, 9eqtrdi 2852 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11 simpl 486 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
1211adantr 484 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Comp)
1312, 1syl 17 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Top)
14 filelss 22461 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) → 𝑥𝑋)
1514adantll 713 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝑋)
162clscld 21656 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
1713, 15, 16syl2anc 587 . . . . 5 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
1817fmpttd 6860 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
1918frnd 6498 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
20 simpr 488 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
2120adantr 484 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐹 ∈ (Fil‘𝑋))
22 simpr 488 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝐹)
232clsss3 21668 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
2413, 15, 23syl2anc 587 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
252sscls 21665 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
2613, 15, 25syl2anc 587 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
27 filss 22462 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹 ∧ ((cls‘𝐽)‘𝑥) ⊆ 𝑋𝑥 ⊆ ((cls‘𝐽)‘𝑥))) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
2821, 22, 24, 26, 27syl13anc 1369 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ∈ 𝐹)
2928fmpttd 6860 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹𝐹)
3029frnd 6498 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
31 fiss 8876 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
3220, 30, 31syl2anc 587 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
33 filfi 22468 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
3420, 33syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘𝐹) = 𝐹)
3532, 34sseqtrd 3958 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
36 0nelfil 22458 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
3720, 36syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
3835, 37ssneldd 3921 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))))
39 cmpfii 22018 . . 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 1368 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ≠ ∅)
4110, 40eqnetrd 3057 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  wss 3884  c0 4246  ifcif 4428   cuni 4803   cint 4841   ciin 4885  cmpt 5113  ran crn 5524  cfv 6328  (class class class)co 7139  ficfi 8862  Topctop 21502  Clsdccld 21625  clsccl 21627  Compccmp 21995  Filcfil 22454   fClus cfcls 22545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-fbas 20092  df-top 21503  df-cld 21628  df-cls 21630  df-cmp 21996  df-fil 22455  df-fcls 22550
This theorem is referenced by:  fclscmp  22639  ufilcmp  22641  relcmpcmet  23926
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