Theorem locfinnei 23691

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 2762	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
3	1, 2	islocfin 23685	. . 3 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
4	3	simp3bi 1164	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
5		eleq1 2850	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ 𝑛 ↔ 𝑃 ∈ 𝑛))
6	5	anbi1d 642	. . . 4 ⊢ (𝑥 = 𝑃 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
7	6	rexbidv 3188	. . 3 ⊢ (𝑥 = 𝑃 → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
8	7	rspccva 3579	. 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 591	1 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3415   ∩ cin 3903  ∅c0 4285  ∪ cuni 4871  ‘cfv 6536  Fincfn 8941  Topctop 23061  LocFinclocfin 23672

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fv 6544  df-top 23062  df-locfin 23675

This theorem is used by:  lfinpfin  23692  lfinun  23693  locfincmp  23694  locfincf  23699