Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pcmplfin Structured version   Visualization version   GIF version

Theorem pcmplfin 30525
 Description: Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement 𝑣 that is locally finite. (Contributed by Thierry Arnoux, 31-Jan-2020.)
Hypothesis
Ref Expression
pcmplfin.x 𝑋 = 𝐽
Assertion
Ref Expression
pcmplfin ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
Distinct variable groups:   𝑣,𝐽   𝑣,𝑈
Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem pcmplfin
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 simp2 1128 . . . 4 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈𝐽)
2 ssexg 5041 . . . . . . 7 ((𝑈𝐽𝐽 ∈ Paracomp) → 𝑈 ∈ V)
32ancoms 452 . . . . . 6 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽) → 𝑈 ∈ V)
433adant3 1123 . . . . 5 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈 ∈ V)
5 elpwg 4387 . . . . 5 (𝑈 ∈ V → (𝑈 ∈ 𝒫 𝐽𝑈𝐽))
64, 5syl 17 . . . 4 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → (𝑈 ∈ 𝒫 𝐽𝑈𝐽))
71, 6mpbird 249 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈 ∈ 𝒫 𝐽)
8 ispcmp 30522 . . . . . 6 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
9 pcmplfin.x . . . . . . 7 𝑋 = 𝐽
109iscref 30509 . . . . . 6 (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
118, 10bitri 267 . . . . 5 (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
1211simprbi 492 . . . 4 (𝐽 ∈ Paracomp → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
13123ad2ant1 1124 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
14 simp3 1129 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑋 = 𝑈)
15 unieq 4679 . . . . . 6 (𝑢 = 𝑈 𝑢 = 𝑈)
1615eqeq2d 2788 . . . . 5 (𝑢 = 𝑈 → (𝑋 = 𝑢𝑋 = 𝑈))
17 breq2 4890 . . . . . 6 (𝑢 = 𝑈 → (𝑣Ref𝑢𝑣Ref𝑈))
1817rexbidv 3237 . . . . 5 (𝑢 = 𝑈 → (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢 ↔ ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈))
1916, 18imbi12d 336 . . . 4 (𝑢 = 𝑈 → ((𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) ↔ (𝑋 = 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
2019rspcv 3507 . . 3 (𝑈 ∈ 𝒫 𝐽 → (∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) → (𝑋 = 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
217, 13, 14, 20syl3c 66 . 2 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)
22 elin 4019 . . . . 5 (𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ↔ (𝑣 ∈ 𝒫 𝐽𝑣 ∈ (LocFin‘𝐽)))
2322anbi1i 617 . . . 4 ((𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈) ↔ ((𝑣 ∈ 𝒫 𝐽𝑣 ∈ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈))
24 anass 462 . . . 4 (((𝑣 ∈ 𝒫 𝐽𝑣 ∈ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈) ↔ (𝑣 ∈ 𝒫 𝐽 ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)))
2523, 24bitri 267 . . 3 ((𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈) ↔ (𝑣 ∈ 𝒫 𝐽 ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)))
2625rexbii2 3222 . 2 (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈 ↔ ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
2721, 26sylib 210 1 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  Vcvv 3398   ∩ cin 3791   ⊆ wss 3792  𝒫 cpw 4379  ∪ cuni 4671   class class class wbr 4886  ‘cfv 6135  Topctop 21105  Refcref 21714  LocFinclocfin 21716  CovHasRefccref 30507  Paracompcpcmp 30520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-cref 30508  df-pcmp 30521 This theorem is referenced by:  pcmplfinf  30526
 Copyright terms: Public domain W3C validator