Theorem locfincf 23761

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 23142	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
2	1	ad2antrr 739	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3		toponuni 23143	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐾)
4	3	ad2antrr 739	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = ∪ 𝐾)
5		locfincf.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6		eqid 2762	. . . . . . 7 ⊢ ∪ 𝑥 = ∪ 𝑥
7	5, 6	locfinbas 23752	. . . . . 6 ⊢ (𝑥 ∈ (LocFin‘𝐽) → 𝑋 = ∪ 𝑥)
8	7	adantl 487	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = ∪ 𝑥)
9	4, 8	eqtr3d 2799	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∪ 𝐾 = ∪ 𝑥)
10	4	eleq2d 2848	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐾))
11	5	locfinnei 23753	. . . . . . . 8 ⊢ ((𝑥 ∈ (LocFin‘𝐽) ∧ 𝑦 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
12	11	ex 418	. . . . . . 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 487	. . . . . . 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 518	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 ∈ 𝑋 → ∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
16	10, 15	sylbird 263	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 ∈ ∪ 𝐾 → ∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
17	16	ralrimiv 3155	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 ∈ ∪ 𝐾∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
18		eqid 2762	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
19	18, 6	islocfin 23747	. . . 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 1362	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑥 ∈ (LocFin‘𝐾))
21	20	ex 418	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ∪ cuni 4870  ‘cfv 6537  Fincfn 8955  Topctop 23122  TopOnctopon 23139  LocFinclocfin 23734

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-top 23123  df-topon 23140  df-locfin 23737

This theorem is used by: (None)