Theorem locfincf 23540

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) ∧ 𝐽 ⊆ 𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))

Proof of Theorem locfincf

Dummy variables 𝑛 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 22920	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
2	1	ad2antrr 726	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3		toponuni 22921	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐾)
4	3	ad2antrr 726	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = ∪ 𝐾)
5		locfincf.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6		eqid 2736	. . . . . . 7 ⊢ ∪ 𝑥 = ∪ 𝑥
7	5, 6	locfinbas 23531	. . . . . 6 ⊢ (𝑥 ∈ (LocFin‘𝐽) → 𝑋 = ∪ 𝑥)
8	7	adantl 481	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = ∪ 𝑥)
9	4, 8	eqtr3d 2778	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∪ 𝐾 = ∪ 𝑥)
10	4	eleq2d 2826	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐾))
11	5	locfinnei 23532	. . . . . . . 8 ⊢ ((𝑥 ∈ (LocFin‘𝐽) ∧ 𝑦 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
12	11	ex 412	. . . . . . 7 ⊢ (𝑥 ∈ (LocFin‘𝐽) → (𝑦 ∈ 𝑋 → ∃𝑛 ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
13		ssrexv 4052	. . . . . . . 8 ⊢ (𝐽 ⊆ 𝐾 → (∃𝑛 ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → ∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
14	13	adantl 481	. . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (∃𝑛 ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) → ∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
15	12, 14	sylan9r 508	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 ∈ 𝑋 → ∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
16	10, 15	sylbird 260	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 ∈ ∪ 𝐾 → ∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
17	16	ralrimiv 3144	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 ∈ ∪ 𝐾∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
18		eqid 2736	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
19	18, 6	islocfin 23526	. . . 4 ⊢ (𝑥 ∈ (LocFin‘𝐾) ↔ (𝐾 ∈ Top ∧ ∪ 𝐾 = ∪ 𝑥 ∧ ∀𝑦 ∈ ∪ 𝐾∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
20	2, 9, 17, 19	syl3anbrc 1343	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑥 ∈ (LocFin‘𝐾))
21	20	ex 412	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (LocFin‘𝐽) → 𝑥 ∈ (LocFin‘𝐾)))
22	21	ssrdv 3988	1 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3435   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  ∪ cuni 4906  ‘cfv 6560  Fincfn 8986  Topctop 22900  TopOnctopon 22917  LocFinclocfin 23513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-top 22901  df-topon 22918  df-locfin 23516

This theorem is referenced by: (None)