MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  locfincf Structured version   Visualization version   GIF version

Theorem locfincf 23042
Description: A locally finite cover in a coarser topology is locally finite in a finer topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
locfincf.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
locfincf ((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (LocFinβ€˜π½) βŠ† (LocFinβ€˜πΎ))

Proof of Theorem locfincf
Dummy variables 𝑛 𝑠 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 22422 . . . . 5 (𝐾 ∈ (TopOnβ€˜π‘‹) β†’ 𝐾 ∈ Top)
21ad2antrr 724 . . . 4 (((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (LocFinβ€˜π½)) β†’ 𝐾 ∈ Top)
3 toponuni 22423 . . . . . 6 (𝐾 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐾)
43ad2antrr 724 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (LocFinβ€˜π½)) β†’ 𝑋 = βˆͺ 𝐾)
5 locfincf.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
6 eqid 2732 . . . . . . 7 βˆͺ π‘₯ = βˆͺ π‘₯
75, 6locfinbas 23033 . . . . . 6 (π‘₯ ∈ (LocFinβ€˜π½) β†’ 𝑋 = βˆͺ π‘₯)
87adantl 482 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (LocFinβ€˜π½)) β†’ 𝑋 = βˆͺ π‘₯)
94, 8eqtr3d 2774 . . . 4 (((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (LocFinβ€˜π½)) β†’ βˆͺ 𝐾 = βˆͺ π‘₯)
104eleq2d 2819 . . . . . 6 (((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (LocFinβ€˜π½)) β†’ (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ βˆͺ 𝐾))
115locfinnei 23034 . . . . . . . 8 ((π‘₯ ∈ (LocFinβ€˜π½) ∧ 𝑦 ∈ 𝑋) β†’ βˆƒπ‘› ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
1211ex 413 . . . . . . 7 (π‘₯ ∈ (LocFinβ€˜π½) β†’ (𝑦 ∈ 𝑋 β†’ βˆƒπ‘› ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
13 ssrexv 4051 . . . . . . . 8 (𝐽 βŠ† 𝐾 β†’ (βˆƒπ‘› ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) β†’ βˆƒπ‘› ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
1413adantl 482 . . . . . . 7 ((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (βˆƒπ‘› ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin) β†’ βˆƒπ‘› ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
1512, 14sylan9r 509 . . . . . 6 (((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (LocFinβ€˜π½)) β†’ (𝑦 ∈ 𝑋 β†’ βˆƒπ‘› ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
1610, 15sylbird 259 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (LocFinβ€˜π½)) β†’ (𝑦 ∈ βˆͺ 𝐾 β†’ βˆƒπ‘› ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
1716ralrimiv 3145 . . . 4 (((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (LocFinβ€˜π½)) β†’ βˆ€π‘¦ ∈ βˆͺ πΎβˆƒπ‘› ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin))
18 eqid 2732 . . . . 5 βˆͺ 𝐾 = βˆͺ 𝐾
1918, 6islocfin 23028 . . . 4 (π‘₯ ∈ (LocFinβ€˜πΎ) ↔ (𝐾 ∈ Top ∧ βˆͺ 𝐾 = βˆͺ π‘₯ ∧ βˆ€π‘¦ ∈ βˆͺ πΎβˆƒπ‘› ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ π‘₯ ∣ (𝑠 ∩ 𝑛) β‰  βˆ…} ∈ Fin)))
202, 9, 17, 19syl3anbrc 1343 . . 3 (((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (LocFinβ€˜π½)) β†’ π‘₯ ∈ (LocFinβ€˜πΎ))
2120ex 413 . 2 ((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (π‘₯ ∈ (LocFinβ€˜π½) β†’ π‘₯ ∈ (LocFinβ€˜πΎ)))
2221ssrdv 3988 1 ((𝐾 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (LocFinβ€˜π½) βŠ† (LocFinβ€˜πΎ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  βˆͺ cuni 4908  β€˜cfv 6543  Fincfn 8941  Topctop 22402  TopOnctopon 22419  LocFinclocfin 23015
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-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-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-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-iota 6495  df-fun 6545  df-fv 6551  df-top 22403  df-topon 22420  df-locfin 23018
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator