Theorem locfinnei 23455

Description: A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover. (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypothesis

Ref	Expression
locfinnei.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
locfinnei	⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))

Distinct variable groups:   𝑛,𝑠,𝐴   𝑛,𝐽   𝑃,𝑛

Allowed substitution hints:   𝑃(𝑠)   𝐽(𝑠)   𝑋(𝑛,𝑠)

Proof of Theorem locfinnei

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		locfinnei.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2		eqid 2728	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
3	1, 2	islocfin 23449	. . 3 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
4	3	simp3bi 1144	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
5		eleq1 2817	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ 𝑛 ↔ 𝑃 ∈ 𝑛))
6	5	anbi1d 629	. . . 4 ⊢ (𝑥 = 𝑃 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
7	6	rexbidv 3176	. . 3 ⊢ (𝑥 = 𝑃 → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
8	7	rspccva 3610	. 2 ⊢ ((∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ∧ 𝑃 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
9	4, 8	sylan 578	1 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  {crab 3430   ∩ cin 3948  ∅c0 4326  ∪ cuni 4912  ‘cfv 6553  Fincfn 8972  Topctop 22823  LocFinclocfin 23436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-top 22824  df-locfin 23439

This theorem is referenced by:  lfinpfin  23456  lfinun  23457  locfincmp  23458  locfincf  23463