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Theorem fclscmpi 23403
Description: Forward direction of fclscmp 23404. 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 22769 . . . 4 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
2 flimfnfcls.x . . . . . 6 𝑋 = βˆͺ 𝐽
32fclsval 23382 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) = if(𝑋 = 𝑋, ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯), βˆ…))
4 eqid 2733 . . . . . 6 𝑋 = 𝑋
54iftruei 4497 . . . . 5 if(𝑋 = 𝑋, ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯), βˆ…) = ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯)
63, 5eqtrdi 2789 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) = ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯))
71, 6sylan 581 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) = ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯))
8 fvex 6859 . . . 4 ((clsβ€˜π½)β€˜π‘₯) ∈ V
98dfiin3 5926 . . 3 ∩ π‘₯ ∈ 𝐹 ((clsβ€˜π½)β€˜π‘₯) = ∩ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯))
107, 9eqtrdi 2789 . 2 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐽 fClus 𝐹) = ∩ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)))
11 simpl 484 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐽 ∈ Comp)
1211adantr 482 . . . . . . 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 23226 . . . . . . 7 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ π‘₯ ∈ 𝐹) β†’ π‘₯ βŠ† 𝑋)
1514adantll 713 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ π‘₯ βŠ† 𝑋)
162clscld 22421 . . . . . 6 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘₯) ∈ (Clsdβ€˜π½))
1713, 15, 16syl2anc 585 . . . . 5 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘₯) ∈ (Clsdβ€˜π½))
1817fmpttd 7067 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)):𝐹⟢(Clsdβ€˜π½))
1918frnd 6680 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)) βŠ† (Clsdβ€˜π½))
20 simpr 486 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
2120adantr 482 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
22 simpr 486 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ π‘₯ ∈ 𝐹)
232clsss3 22433 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜π‘₯) βŠ† 𝑋)
2413, 15, 23syl2anc 585 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘₯) βŠ† 𝑋)
252sscls 22430 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ π‘₯ βŠ† 𝑋) β†’ π‘₯ βŠ† ((clsβ€˜π½)β€˜π‘₯))
2613, 15, 25syl2anc 585 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ π‘₯ βŠ† ((clsβ€˜π½)β€˜π‘₯))
27 filss 23227 . . . . . . . . 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 1373 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ π‘₯ ∈ 𝐹) β†’ ((clsβ€˜π½)β€˜π‘₯) ∈ 𝐹)
2928fmpttd 7067 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)):𝐹⟢𝐹)
3029frnd 6680 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯)) βŠ† 𝐹)
31 fiss 9368 . . . . . 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 585 . . . . 5 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (fiβ€˜ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯))) βŠ† (fiβ€˜πΉ))
33 filfi 23233 . . . . . 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 3988 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (fiβ€˜ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯))) βŠ† 𝐹)
36 0nelfil 23223 . . . . 5 (𝐹 ∈ (Filβ€˜π‘‹) β†’ Β¬ βˆ… ∈ 𝐹)
3720, 36syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ Β¬ βˆ… ∈ 𝐹)
3835, 37ssneldd 3951 . . 3 ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ Β¬ βˆ… ∈ (fiβ€˜ran (π‘₯ ∈ 𝐹 ↦ ((clsβ€˜π½)β€˜π‘₯))))
39 cmpfii 22783 . . 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 1372 . 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   βŠ† wss 3914  βˆ…c0 4286  ifcif 4490  βˆͺ cuni 4869  βˆ© cint 4911  βˆ© ciin 4959   ↦ cmpt 5192  ran crn 5638  β€˜cfv 6500  (class class class)co 7361  ficfi 9354  Topctop 22265  Clsdccld 22390  clsccl 22392  Compccmp 22760  Filcfil 23219   fClus cfcls 23310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1o 8416  df-er 8654  df-en 8890  df-fin 8893  df-fi 9355  df-fbas 20816  df-top 22266  df-cld 22393  df-cls 22395  df-cmp 22761  df-fil 23220  df-fcls 23315
This theorem is referenced by:  fclscmp  23404  ufilcmp  23406  relcmpcmet  24705
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