Theorem locfincf 23479

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 22861	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
2	1	ad2antrr 727	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3		toponuni 22862	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐾)
4	3	ad2antrr 727	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = ∪ 𝐾)
5		locfincf.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6		eqid 2737	. . . . . . 7 ⊢ ∪ 𝑥 = ∪ 𝑥
7	5, 6	locfinbas 23470	. . . . . 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 2774	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∪ 𝐾 = ∪ 𝑥)
10	4	eleq2d 2823	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐾))
11	5	locfinnei 23471	. . . . . . . 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 4004	. . . . . . . 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 3128	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 ∈ ∪ 𝐾∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
18		eqid 2737	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
19	18, 6	islocfin 23465	. . . 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 1345	. . 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 3940	1 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3400   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  ∪ cuni 4864  ‘cfv 6493  Fincfn 8887  Topctop 22841  TopOnctopon 22858  LocFinclocfin 23452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fv 6501  df-top 22842  df-topon 22859  df-locfin 23455

This theorem is referenced by: (None)