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Theorem locfinbas 22125
 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.
StepHypRef Expression
1 locfinbas.1 . . 3 𝑋 = 𝐽
2 locfinbas.2 . . 3 𝑌 = 𝐴
31, 2islocfin 22120 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠𝑋𝑛𝐽 (𝑠𝑛 ∧ {𝑥𝐴 ∣ (𝑥𝑛) ≠ ∅} ∈ Fin)))
43simp2bi 1143 1 (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ∀wral 3130  ∃wrex 3131  {crab 3134   ∩ cin 3907  ∅c0 4265  ∪ cuni 4813  ‘cfv 6334  Fincfn 8496  Topctop 21496  LocFinclocfin 22107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342  df-top 21497  df-locfin 22110 This theorem is referenced by:  lfinpfin  22127  lfinun  22128  locfincmp  22129  locfindis  22133  locfincf  22134
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