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Theorem locfincf 21144
 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 20541 . . . . 5 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
21ad2antrr 758 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3 toponuni 20542 . . . . . 6 (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = 𝐾)
43ad2antrr 758 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = 𝐾)
5 locfincf.1 . . . . . . 7 𝑋 = 𝐽
6 eqid 2610 . . . . . . 7 𝑥 = 𝑥
75, 6locfinbas 21135 . . . . . 6 (𝑥 ∈ (LocFin‘𝐽) → 𝑋 = 𝑥)
87adantl 481 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑥)
94, 8eqtr3d 2646 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 = 𝑥)
104eleq2d 2673 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋𝑦 𝐾))
115locfinnei 21136 . . . . . . . 8 ((𝑥 ∈ (LocFin‘𝐽) ∧ 𝑦𝑋) → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
1211ex 449 . . . . . . 7 (𝑥 ∈ (LocFin‘𝐽) → (𝑦𝑋 → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
13 ssrexv 3630 . . . . . . . 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 688 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1610, 15sylbird 249 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 𝐾 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1716ralrimiv 2948 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
18 eqid 2610 . . . . 5 𝐾 = 𝐾
1918, 6islocfin 21130 . . . 4 (𝑥 ∈ (LocFin‘𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝑥 ∧ ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
202, 9, 17, 19syl3anbrc 1239 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑥 ∈ (LocFin‘𝐾))
2120ex 449 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (LocFin‘𝐽) → 𝑥 ∈ (LocFin‘𝐾)))
2221ssrdv 3574 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ‘cfv 5804  Fincfn 7841  Topctop 20517  TopOnctopon 20518  LocFinclocfin 21117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-topon 20523  df-locfin 21120 This theorem is referenced by: (None)
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