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Theorem fclsss2 23747
Description: A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
fclsss2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) β†’ (𝐽 fClus 𝐺) βŠ† (𝐽 fClus 𝐹))
Proof of Theorem fclsss2
Dummy variables 𝑠 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1191 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) ∧ π‘₯ ∈ (𝐽 fClus 𝐺)) β†’ 𝐹 βŠ† 𝐺)
2 ssralv 4049 . . . . . 6 (𝐹 βŠ† 𝐺 β†’ (βˆ€π‘  ∈ 𝐺 π‘₯ ∈ ((clsβ€˜π½)β€˜π‘ ) β†’ βˆ€π‘  ∈ 𝐹 π‘₯ ∈ ((clsβ€˜π½)β€˜π‘ )))
31, 2syl 17 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) ∧ π‘₯ ∈ (𝐽 fClus 𝐺)) β†’ (βˆ€π‘  ∈ 𝐺 π‘₯ ∈ ((clsβ€˜π½)β€˜π‘ ) β†’ βˆ€π‘  ∈ 𝐹 π‘₯ ∈ ((clsβ€˜π½)β€˜π‘ )))
4 simpl1 1189 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) ∧ π‘₯ ∈ (𝐽 fClus 𝐺)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
5 fclstopon 23736 . . . . . . . 8 (π‘₯ ∈ (𝐽 fClus 𝐺) β†’ (𝐽 ∈ (TopOnβ€˜π‘‹) ↔ 𝐺 ∈ (Filβ€˜π‘‹)))
65adantl 480 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) ∧ π‘₯ ∈ (𝐽 fClus 𝐺)) β†’ (𝐽 ∈ (TopOnβ€˜π‘‹) ↔ 𝐺 ∈ (Filβ€˜π‘‹)))
74, 6mpbid 231 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) ∧ π‘₯ ∈ (𝐽 fClus 𝐺)) β†’ 𝐺 ∈ (Filβ€˜π‘‹))
8 isfcls2 23737 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐺 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ (𝐽 fClus 𝐺) ↔ βˆ€π‘  ∈ 𝐺 π‘₯ ∈ ((clsβ€˜π½)β€˜π‘ )))
94, 7, 8syl2anc 582 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) ∧ π‘₯ ∈ (𝐽 fClus 𝐺)) β†’ (π‘₯ ∈ (𝐽 fClus 𝐺) ↔ βˆ€π‘  ∈ 𝐺 π‘₯ ∈ ((clsβ€˜π½)β€˜π‘ )))
10 simpl2 1190 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) ∧ π‘₯ ∈ (𝐽 fClus 𝐺)) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
11 isfcls2 23737 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ (𝐽 fClus 𝐹) ↔ βˆ€π‘  ∈ 𝐹 π‘₯ ∈ ((clsβ€˜π½)β€˜π‘ )))
124, 10, 11syl2anc 582 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) ∧ π‘₯ ∈ (𝐽 fClus 𝐺)) β†’ (π‘₯ ∈ (𝐽 fClus 𝐹) ↔ βˆ€π‘  ∈ 𝐹 π‘₯ ∈ ((clsβ€˜π½)β€˜π‘ )))
133, 9, 123imtr4d 293 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) ∧ π‘₯ ∈ (𝐽 fClus 𝐺)) β†’ (π‘₯ ∈ (𝐽 fClus 𝐺) β†’ π‘₯ ∈ (𝐽 fClus 𝐹)))
1413ex 411 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) β†’ (π‘₯ ∈ (𝐽 fClus 𝐺) β†’ (π‘₯ ∈ (𝐽 fClus 𝐺) β†’ π‘₯ ∈ (𝐽 fClus 𝐹))))
1514pm2.43d 53 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) β†’ (π‘₯ ∈ (𝐽 fClus 𝐺) β†’ π‘₯ ∈ (𝐽 fClus 𝐹)))
1615ssrdv 3987 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐹 βŠ† 𝐺) β†’ (𝐽 fClus 𝐺) βŠ† (𝐽 fClus 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   ∈ wcel 2104  βˆ€wral 3059   βŠ† wss 3947  β€˜cfv 6542  (class class class)co 7411  TopOnctopon 22632  clsccl 22742  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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-topon 22633  df-fil 23570  df-fcls 23665
This theorem is referenced by:  fclsfnflim  23751  ufilcmp  23756  cnpfcfi  23764
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