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Theorem fclscmpi 23977
Description: Forward direction of fclscmp 23978. 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 23343 . . . 4 (𝐽 ∈ Comp → 𝐽 ∈ Top)
2 flimfnfcls.x . . . . . 6 𝑋 = 𝐽
32fclsval 23956 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅))
4 eqid 2725 . . . . . 6 𝑋 = 𝑋
54iftruei 4537 . . . . 5 if(𝑋 = 𝑋, 𝑥𝐹 ((cls‘𝐽)‘𝑥), ∅) = 𝑥𝐹 ((cls‘𝐽)‘𝑥)
63, 5eqtrdi 2781 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
71, 6sylan 578 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = 𝑥𝐹 ((cls‘𝐽)‘𝑥))
8 fvex 6909 . . . 4 ((cls‘𝐽)‘𝑥) ∈ V
98dfiin3 5970 . . 3 𝑥𝐹 ((cls‘𝐽)‘𝑥) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))
107, 9eqtrdi 2781 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) = ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)))
11 simpl 481 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐽 ∈ Comp)
1211adantr 479 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Comp)
1312, 1syl 17 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐽 ∈ Top)
14 filelss 23800 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹) → 𝑥𝑋)
1514adantll 712 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝑋)
162clscld 22995 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
1713, 15, 16syl2anc 582 . . . . 5 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
1817fmpttd 7124 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹⟶(Clsd‘𝐽))
1918frnd 6731 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ (Clsd‘𝐽))
20 simpr 483 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
2120adantr 479 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝐹 ∈ (Fil‘𝑋))
22 simpr 483 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥𝐹)
232clsss3 23007 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
2413, 15, 23syl2anc 582 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → ((cls‘𝐽)‘𝑥) ⊆ 𝑋)
252sscls 23004 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝑋) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
2613, 15, 25syl2anc 582 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥𝐹) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
27 filss 23801 . . . . . . . . 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 7124 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)):𝐹𝐹)
3029frnd 6731 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹)
31 fiss 9449 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥)) ⊆ 𝐹) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
3220, 30, 31syl2anc 582 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ (fi‘𝐹))
33 filfi 23807 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
3420, 33syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘𝐹) = 𝐹)
3532, 34sseqtrd 4017 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))) ⊆ 𝐹)
36 0nelfil 23797 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
3720, 36syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
3835, 37ssneldd 3979 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (fi‘ran (𝑥𝐹 ↦ ((cls‘𝐽)‘𝑥))))
39 cmpfii 23357 . . 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 2997 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2929  wss 3944  c0 4322  ifcif 4530   cuni 4909   cint 4950   ciin 4998  cmpt 5232  ran crn 5679  cfv 6549  (class class class)co 7419  ficfi 9435  Topctop 22839  Clsdccld 22964  clsccl 22966  Compccmp 23334  Filcfil 23793   fClus cfcls 23884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1o 8487  df-er 8725  df-en 8965  df-fin 8968  df-fi 9436  df-fbas 21293  df-top 22840  df-cld 22967  df-cls 22969  df-cmp 23335  df-fil 23794  df-fcls 23889
This theorem is referenced by:  fclscmp  23978  ufilcmp  23980  relcmpcmet  25290
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