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Theorem locfincf 21555
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 20938 . . . . 5 (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
21ad2antrr 705 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3 toponuni 20939 . . . . . 6 (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = 𝐾)
43ad2antrr 705 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = 𝐾)
5 locfincf.1 . . . . . . 7 𝑋 = 𝐽
6 eqid 2771 . . . . . . 7 𝑥 = 𝑥
75, 6locfinbas 21546 . . . . . 6 (𝑥 ∈ (LocFin‘𝐽) → 𝑋 = 𝑥)
87adantl 467 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = 𝑥)
94, 8eqtr3d 2807 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 = 𝑥)
104eleq2d 2836 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋𝑦 𝐾))
115locfinnei 21547 . . . . . . . 8 ((𝑥 ∈ (LocFin‘𝐽) ∧ 𝑦𝑋) → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
1211ex 397 . . . . . . 7 (𝑥 ∈ (LocFin‘𝐽) → (𝑦𝑋 → ∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
13 ssrexv 3816 . . . . . . . 8 (𝐽𝐾 → (∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1413adantl 467 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (∃𝑛𝐽 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1512, 14sylan9r 498 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦𝑋 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1610, 15sylbird 250 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 𝐾 → ∃𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
1716ralrimiv 3114 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
18 eqid 2771 . . . . 5 𝐾 = 𝐾
1918, 6islocfin 21541 . . . 4 (𝑥 ∈ (LocFin‘𝐾) ↔ (𝐾 ∈ Top ∧ 𝐾 = 𝑥 ∧ ∀𝑦 𝐾𝑛𝐾 (𝑦𝑛 ∧ {𝑠𝑥 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
202, 9, 17, 19syl3anbrc 1428 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑥 ∈ (LocFin‘𝐾))
2120ex 397 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (LocFin‘𝐽) → 𝑥 ∈ (LocFin‘𝐾)))
2221ssrdv 3758 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  cin 3722  wss 3723  c0 4063   cuni 4575  cfv 6030  Fincfn 8113  Topctop 20918  TopOnctopon 20935  LocFinclocfin 21528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-top 20919  df-topon 20936  df-locfin 21531
This theorem is referenced by: (None)
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