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Theorem locfincf 22428
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 21810 . . . . 5 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
21ad2antrr 726 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3 toponuni 21811 . . . . . 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 2737 . . . . . . 7 𝑥 = 𝑥
75, 6locfinbas 22419 . . . . . 6 (𝑥 ∈ (LocFin‘𝐽) → 𝑋 = 𝑥)
87adantl 485 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑥)
94, 8eqtr3d 2779 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 = 𝑥)
104eleq2d 2823 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋𝑦 𝐾))
115locfinnei 22420 . . . . . . . 8 ((𝑥 ∈ (LocFin‘𝐽) ∧ 𝑦𝑋) → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
1211ex 416 . . . . . . 7 (𝑥 ∈ (LocFin‘𝐽) → (𝑦𝑋 → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
13 ssrexv 3968 . . . . . . . 8 (𝐽𝐾 → (∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1413adantl 485 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1512, 14sylan9r 512 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1610, 15sylbird 263 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 𝐾 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1716ralrimiv 3104 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
18 eqid 2737 . . . . 5 𝐾 = 𝐾
1918, 6islocfin 22414 . . . 4 (𝑥 ∈ (LocFin‘𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝑥 ∧ ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
202, 9, 17, 19syl3anbrc 1345 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑥 ∈ (LocFin‘𝐾))
2120ex 416 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (LocFin‘𝐽) → 𝑥 ∈ (LocFin‘𝐾)))
2221ssrdv 3907 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wne 2940  wral 3061  wrex 3062  {crab 3065  cin 3865  wss 3866  c0 4237   cuni 4819  cfv 6380  Fincfn 8626  Topctop 21790  TopOnctopon 21807  LocFinclocfin 22401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fv 6388  df-top 21791  df-topon 21808  df-locfin 22404
This theorem is referenced by: (None)
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