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Theorem locfincf 23555
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 22935 . . . . 5 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
21ad2antrr 726 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3 toponuni 22936 . . . . . 6 (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = 𝐾)
43ad2antrr 726 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = 𝐾)
5 locfincf.1 . . . . . . 7 𝑋 = 𝐽
6 eqid 2735 . . . . . . 7 𝑥 = 𝑥
75, 6locfinbas 23546 . . . . . 6 (𝑥 ∈ (LocFin‘𝐽) → 𝑋 = 𝑥)
87adantl 481 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑥)
94, 8eqtr3d 2777 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 = 𝑥)
104eleq2d 2825 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋𝑦 𝐾))
115locfinnei 23547 . . . . . . . 8 ((𝑥 ∈ (LocFin‘𝐽) ∧ 𝑦𝑋) → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
1211ex 412 . . . . . . 7 (𝑥 ∈ (LocFin‘𝐽) → (𝑦𝑋 → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
13 ssrexv 4065 . . . . . . . 8 (𝐽𝐾 → (∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1413adantl 481 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1512, 14sylan9r 508 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1610, 15sylbird 260 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 𝐾 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1716ralrimiv 3143 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
18 eqid 2735 . . . . 5 𝐾 = 𝐾
1918, 6islocfin 23541 . . . 4 (𝑥 ∈ (LocFin‘𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝑥 ∧ ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
202, 9, 17, 19syl3anbrc 1342 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑥 ∈ (LocFin‘𝐾))
2120ex 412 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (LocFin‘𝐽) → 𝑥 ∈ (LocFin‘𝐾)))
2221ssrdv 4001 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  cin 3962  wss 3963  c0 4339   cuni 4912  cfv 6563  Fincfn 8984  Topctop 22915  TopOnctopon 22932  LocFinclocfin 23528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-top 22916  df-topon 22933  df-locfin 23531
This theorem is referenced by: (None)
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