Theorem locfincf 23812

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 23193	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
2	1	ad2antrr 739	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3		toponuni 23194	. . . . . 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 2760	. . . . . . 7 ⊢ ∪ 𝑥 = ∪ 𝑥
7	5, 6	locfinbas 23803	. . . . . 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 2797	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∪ 𝐾 = ∪ 𝑥)
10	4	eleq2d 2846	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐾))
11	5	locfinnei 23804	. . . . . . . 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 4000	. . . . . . . 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 3153	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 ∈ ∪ 𝐾∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
18		eqid 2760	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
19	18, 6	islocfin 23798	. . . 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 3936	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 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  ∪ cuni 4866  ‘cfv 6527  Fincfn 8951  Topctop 23173  TopOnctopon 23190  LocFinclocfin 23785

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 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fv 6535  df-top 23174  df-topon 23191  df-locfin 23788

This theorem is used by: (None)