Theorem locfinbas 23475

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 23470	. 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑠 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑠 ∈ 𝑛 ∧ {𝑥 ∈ 𝐴 ∣ (𝑥 ∩ 𝑛) ≠ ∅} ∈ Fin)))
4	3	simp2bi 1147	1 ⊢ (𝐴 ∈ (LocFin‘𝐽) → 𝑋 = 𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2930  ∀wral 3049  ∃wrex 3059  {crab 3387   ∩ cin 3884  ∅c0 4263  ∪ cuni 4840  ‘cfv 6487  Fincfn 8882  Topctop 22846  LocFinclocfin 23457

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fv 6495  df-top 22847  df-locfin 23460

This theorem is referenced by:  lfinpfin  23477  lfinun  23478  locfincmp  23479  locfindis  23483  locfincf  23484