Theorem locfinbas 23648

Description: A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypotheses

Ref	Expression
locfinbas.1	⊢ 𝑋 = ∪ 𝐽
locfinbas.2	⊢ 𝑌 = ∪ 𝐴

Assertion

Ref	Expression
locfinbas	⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)

Proof of Theorem locfinbas

Dummy variables 𝑛 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		locfinbas.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2		locfinbas.2	. . 3 ⊢ 𝑌 = ∪ 𝐴
3	1, 2	islocfin 23643	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {𝑥 ∈ 𝐴 ∣ (𝑥 ∩ 𝑛) ≠ ∅} ∈ Fin)))
4	3	simp2bi 1162	1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3422   ∩ cin 3910  ∅c0 4292  ∪ cuni 4874  ‘cfv 6537  Fincfn 8943  Topctop 23019  LocFinclocfin 23630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-top 23020  df-locfin 23633

This theorem is referenced by:  lfinpfin  23650  lfinun  23651  locfincmp  23652  locfindis  23656  locfincf  23657