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Theorem pcmplfin 31152
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 ssexg 5213 . . . . . 6 ((𝑈𝐽𝐽 ∈ Paracomp) → 𝑈 ∈ V)
21ancoms 462 . . . . 5 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽) → 𝑈 ∈ V)
323adant3 1129 . . . 4 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈 ∈ V)
4 simp2 1134 . . . 4 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈𝐽)
53, 4elpwd 4529 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈 ∈ 𝒫 𝐽)
6 ispcmp 31149 . . . . . 6 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
7 pcmplfin.x . . . . . . 7 𝑋 = 𝐽
87iscref 31136 . . . . . 6 (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
96, 8bitri 278 . . . . 5 (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
109simprbi 500 . . . 4 (𝐽 ∈ Paracomp → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
11103ad2ant1 1130 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
12 simp3 1135 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑋 = 𝑈)
13 unieq 4835 . . . . . 6 (𝑢 = 𝑈 𝑢 = 𝑈)
1413eqeq2d 2835 . . . . 5 (𝑢 = 𝑈 → (𝑋 = 𝑢𝑋 = 𝑈))
15 breq2 5056 . . . . . 6 (𝑢 = 𝑈 → (𝑣Ref𝑢𝑣Ref𝑈))
1615rexbidv 3290 . . . . 5 (𝑢 = 𝑈 → (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢 ↔ ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈))
1714, 16imbi12d 348 . . . 4 (𝑢 = 𝑈 → ((𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) ↔ (𝑋 = 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
1817rspcv 3604 . . 3 (𝑈 ∈ 𝒫 𝐽 → (∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) → (𝑋 = 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
195, 11, 12, 18syl3c 66 . 2 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)
20 rexin 4200 . 2 (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈 ↔ ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
2119, 20sylib 221 1 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134  Vcvv 3480  cin 3918  wss 3919  𝒫 cpw 4521   cuni 4824   class class class wbr 5052  cfv 6343  Topctop 21494  Refcref 22103  LocFinclocfin 22105  CovHasRefccref 31134  Paracompcpcmp 31147
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-cref 31135  df-pcmp 31148
This theorem is referenced by:  pcmplfinf  31153
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