MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fclsss1 Structured version   Visualization version   GIF version
Theorem fclsss1 23746
Description: A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclsss1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (𝐾 fClus 𝐹) βŠ† (𝐽 fClus 𝐹))
Proof of Theorem fclsss1
Dummy variables π‘œ 𝑠 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1191 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fClus 𝐹)) β†’ 𝐽 βŠ† 𝐾)
2 ssralv 4049 . . . . . . 7 (𝐽 βŠ† 𝐾 β†’ (βˆ€π‘œ ∈ 𝐾 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…) β†’ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…)))
32anim2d 610 . . . . . 6 (𝐽 βŠ† 𝐾 β†’ ((π‘₯ ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐾 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…)) β†’ (π‘₯ ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))))
41, 3syl 17 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fClus 𝐹)) β†’ ((π‘₯ ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐾 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…)) β†’ (π‘₯ ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))))
5 simpl2 1190 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fClus 𝐹)) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
6 fclstopon 23736 . . . . . . . 8 (π‘₯ ∈ (𝐾 fClus 𝐹) β†’ (𝐾 ∈ (TopOnβ€˜π‘‹) ↔ 𝐹 ∈ (Filβ€˜π‘‹)))
76adantl 480 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fClus 𝐹)) β†’ (𝐾 ∈ (TopOnβ€˜π‘‹) ↔ 𝐹 ∈ (Filβ€˜π‘‹)))
85, 7mpbird 256 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fClus 𝐹)) β†’ 𝐾 ∈ (TopOnβ€˜π‘‹))
9 fclsopn 23738 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ (𝐾 fClus 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐾 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))))
108, 5, 9syl2anc 582 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fClus 𝐹)) β†’ (π‘₯ ∈ (𝐾 fClus 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐾 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))))
11 simpl1 1189 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fClus 𝐹)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
12 fclsopn 23738 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ (𝐽 fClus 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))))
1311, 5, 12syl2anc 582 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fClus 𝐹)) β†’ (π‘₯ ∈ (𝐽 fClus 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ βˆ€π‘  ∈ 𝐹 (π‘œ ∩ 𝑠) β‰  βˆ…))))
144, 10, 133imtr4d 293 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fClus 𝐹)) β†’ (π‘₯ ∈ (𝐾 fClus 𝐹) β†’ π‘₯ ∈ (𝐽 fClus 𝐹)))
1514ex 411 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (π‘₯ ∈ (𝐾 fClus 𝐹) β†’ (π‘₯ ∈ (𝐾 fClus 𝐹) β†’ π‘₯ ∈ (𝐽 fClus 𝐹))))
1615pm2.43d 53 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (π‘₯ ∈ (𝐾 fClus 𝐹) β†’ π‘₯ ∈ (𝐽 fClus 𝐹)))
1716ssrdv 3987 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (𝐾 fClus 𝐹) βŠ† (𝐽 fClus 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  β€˜cfv 6542  (class class class)co 7411  TopOnctopon 22632  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-rep 5284  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-reu 3375  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-iun 4998  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-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-top 22616  df-topon 22633  df-cld 22743  df-ntr 22744  df-cls 22745  df-fil 23570  df-fcls 23665
This theorem is referenced by:  fclscf  23749
  Copyright terms: Public domain W3C validator