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Theorem fclscmpi 23753
Description: Forward direction of fclscmp 23754. 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 = βˆͺ 𝐽
Assertion
Ref Expression
fclscmpi ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) β‰  βˆ…)

Proof of Theorem fclscmpi
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 cmptop 23119 . . . 4 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
2 flimfnfcls.x . . . . . 6 𝑋 = βˆͺ 𝐽
32fclsval 23732 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯), βˆ…))
4 eqid 2732 . . . . . 6 𝑋 = 𝑋
54iftruei 4535 . . . . 5 if(𝑋 = 𝑋, ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯), βˆ…) = ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯)
63, 5eqtrdi 2788 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) = ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯))
71, 6sylan 580 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) = ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯))
8 fvex 6904 . . . 4 ((clsβ€˜π½)β€˜π‘₯) ∈ V
98dfiin3 5966 . . 3 ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯) = ∩ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯))
107, 9eqtrdi 2788 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) = ∩ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)))
11 simpl 483 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐽 ∈ Comp)
1211adantr 481 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ 𝐽 ∈ Comp)
1312, 1syl 17 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ 𝐽 ∈ Top)
14 filelss 23576 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘₯ ∈ 𝐹) β†’ π‘₯ βŠ† 𝑋)
1514adantll 712 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ π‘₯ βŠ† 𝑋)
162clscld 22771 . . . . . 6 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘₯) ∈ (Clsdβ€˜π½))
1713, 15, 16syl2anc 584 . . . . 5 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘₯) ∈ (Clsdβ€˜π½))
1817fmpttd 7116 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)):𝐹⟢(Clsdβ€˜π½))
1918frnd 6725 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)) βŠ† (Clsdβ€˜π½))
20 simpr 485 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
2120adantr 481 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
22 simpr 485 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ π‘₯ ∈ 𝐹)
232clsss3 22783 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘₯) βŠ† 𝑋)
2413, 15, 23syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘₯) βŠ† 𝑋)
252sscls 22780 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ π‘₯ βŠ† ((clsβ€˜π½)β€˜π‘₯))
2613, 15, 25syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ π‘₯ βŠ† ((clsβ€˜π½)β€˜π‘₯))
27 filss 23577 . . . . . . . . 9 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (π‘₯ ∈ 𝐹 ∧ ((clsβ€˜π½)β€˜π‘₯) βŠ† 𝑋 ∧ π‘₯ βŠ† ((clsβ€˜π½)β€˜π‘₯))) β†’ ((clsβ€˜π½)β€˜π‘₯) ∈ 𝐹)
2821, 22, 24, 26, 27syl13anc 1372 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘₯) ∈ 𝐹)
2928fmpttd 7116 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)):𝐹⟢𝐹)
3029frnd 6725 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)) βŠ† 𝐹)
31 fiss 9421 . . . . . 6 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)) βŠ† 𝐹) β†’ (fiβ€˜ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯))) βŠ† (fiβ€˜πΉ))
3220, 30, 31syl2anc 584 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (fiβ€˜ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯))) βŠ† (fiβ€˜πΉ))
33 filfi 23583 . . . . . 6 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (fiβ€˜πΉ) = 𝐹)
3420, 33syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (fiβ€˜πΉ) = 𝐹)
3532, 34sseqtrd 4022 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (fiβ€˜ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯))) βŠ† 𝐹)
36 0nelfil 23573 . . . . 5 (𝐹 ∈ (Filβ€˜π‘‹) β†’ Β¬ βˆ… ∈ 𝐹)
3720, 36syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ Β¬ βˆ… ∈ 𝐹)
3835, 37ssneldd 3985 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ Β¬ βˆ… ∈ (fiβ€˜ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯))))
39 cmpfii 23133 . . 3 ((𝐽 ∈ Comp ∧ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)) βŠ† (Clsdβ€˜π½) ∧ Β¬ βˆ… ∈ (fiβ€˜ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)))) β†’ ∩ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)) β‰  βˆ…)
4011, 19, 38, 39syl3anc 1371 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ ∩ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)) β‰  βˆ…)
4110, 40eqnetrd 3008 1 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βŠ† wss 3948  βˆ…c0 4322  ifcif 4528  βˆͺ cuni 4908  βˆ© cint 4950  βˆ© ciin 4998   ↦ cmpt 5231  ran crn 5677  β€˜cfv 6543  (class class class)co 7411  ficfi 9407  Topctop 22615  Clsdccld 22740  clsccl 22742  Compccmp 23110  Filcfil 23569   fClus cfcls 23660
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408  df-fbas 21141  df-top 22616  df-cld 22743  df-cls 22745  df-cmp 23111  df-fil 23570  df-fcls 23665
This theorem is referenced by:  fclscmp  23754  ufilcmp  23756  relcmpcmet  25059
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