Theorem locfinbas 23802

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 23797	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {𝑥 ∈ 𝐴 ∣ (𝑥 ∩ 𝑛) ≠ ∅} ∈ Fin)))
4	3	simp2bi 1164	1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ∩ cin 3897  ∅c0 4278  ∪ cuni 4866  ‘cfv 6527  Fincfn 8951  Topctop 23172  LocFinclocfin 23784

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  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 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fv 6535  df-top 23173  df-locfin 23787

This theorem is used by:  lfinpfin  23804  lfinun  23805  locfincmp  23806  locfindis  23810  locfincf  23811