Theorem locfincf 23035

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 22415	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top)
2	1	ad2antrr 725	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝐾 ∈ Top)
3		toponuni 22416	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐾)
4	3	ad2antrr 725	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = ∪ 𝐾)
5		locfincf.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6		eqid 2733	. . . . . . 7 ⊢ ∪ 𝑥 = ∪ 𝑥
7	5, 6	locfinbas 23026	. . . . . 6 ⊢ (𝑥 ∈ (LocFin‘𝐽) → 𝑋 = ∪ 𝑥)
8	7	adantl 483	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑋 = ∪ 𝑥)
9	4, 8	eqtr3d 2775	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∪ 𝐾 = ∪ 𝑥)
10	4	eleq2d 2820	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ ∪ 𝐾))
11	5	locfinnei 23027	. . . . . . . 8 ⊢ ((𝑥 ∈ (LocFin‘𝐽) ∧ 𝑦 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
12	11	ex 414	. . . . . . 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 483	. . . . . . 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 510	. . . . . 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 3146	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → ∀𝑦 ∈ ∪ 𝐾∃𝑛 ∈ 𝐾 (𝑦 ∈ 𝑛 ∧ {𝑠 ∈ 𝑥 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
18		eqid 2733	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
19	18, 6	islocfin 23021	. . . 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 1344	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (LocFin‘𝐽)) → 𝑥 ∈ (LocFin‘𝐾))
21	20	ex 414	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (LocFin‘𝐽) → 𝑥 ∈ (LocFin‘𝐾)))
22	21	ssrdv 3989	1 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (LocFin‘𝐽) ⊆ (LocFin‘𝐾))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {crab 3433   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  ∪ cuni 4909  ‘cfv 6544  Fincfn 8939  Topctop 22395  TopOnctopon 22412  LocFinclocfin 23008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-top 22396  df-topon 22413  df-locfin 23011

This theorem is referenced by: (None)