Theorem locfinnei 23842

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 2760	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
3	1, 2	islocfin 23836	. . 3 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
4	3	simp3bi 1165	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))
5		eleq1 2848	. . . . 5 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ 𝑛 ↔ 𝑃 ∈ 𝑛))
6	5	anbi1d 643	. . . 4 ⊢ (𝑥 = 𝑃 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
7	6	rexbidv 3186	. . 3 ⊢ (𝑥 = 𝑃 → (∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)))
8	7	rspccva 3575	. 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 592	1 ⊢ ((𝐴 ∈ (LocFin‘𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑃 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ∩ cin 4089  ∅c0 4278  ∪ cuni 4866  ‘cfv 6538  Fincfn 8973  Topctop 23211  LocFinclocfin 23823

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-top 23212  df-locfin 23826

This theorem is used by:  lfinpfin  23843  lfinun  23844  locfincmp  23845  locfincf  23850