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Theorem locfincf 22067
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 = 𝐽
Assertion
Ref Expression
locfincf ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))

Proof of Theorem locfincf
Dummy variables 𝑛 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 21449 . . . . 5 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
21ad2antrr 722 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3 toponuni 21450 . . . . . 6 (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = 𝐾)
43ad2antrr 722 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = 𝐾)
5 locfincf.1 . . . . . . 7 𝑋 = 𝐽
6 eqid 2818 . . . . . . 7 𝑥 = 𝑥
75, 6locfinbas 22058 . . . . . 6 (𝑥 ∈ (LocFin‘𝐽) → 𝑋 = 𝑥)
87adantl 482 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑥)
94, 8eqtr3d 2855 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 = 𝑥)
104eleq2d 2895 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋𝑦 𝐾))
115locfinnei 22059 . . . . . . . 8 ((𝑥 ∈ (LocFin‘𝐽) ∧ 𝑦𝑋) → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
1211ex 413 . . . . . . 7 (𝑥 ∈ (LocFin‘𝐽) → (𝑦𝑋 → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
13 ssrexv 4031 . . . . . . . 8 (𝐽𝐾 → (∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1413adantl 482 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1512, 14sylan9r 509 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1610, 15sylbird 261 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 𝐾 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1716ralrimiv 3178 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
18 eqid 2818 . . . . 5 𝐾 = 𝐾
1918, 6islocfin 22053 . . . 4 (𝑥 ∈ (LocFin‘𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝑥 ∧ ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
202, 9, 17, 19syl3anbrc 1335 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑥 ∈ (LocFin‘𝐾))
2120ex 413 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (LocFin‘𝐽) → 𝑥 ∈ (LocFin‘𝐾)))
2221ssrdv 3970 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  cin 3932  wss 3933  c0 4288   cuni 4830  cfv 6348  Fincfn 8497  Topctop 21429  TopOnctopon 21446  LocFinclocfin 22040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21430  df-topon 21447  df-locfin 22043
This theorem is referenced by: (None)
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