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Theorem pcmplfin 33859
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 5323 . . . . . 6 ((𝑈𝐽𝐽 ∈ Paracomp) → 𝑈 ∈ V)
21ancoms 458 . . . . 5 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽) → 𝑈 ∈ V)
323adant3 1133 . . . 4 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈 ∈ V)
4 simp2 1138 . . . 4 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈𝐽)
53, 4elpwd 4606 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈 ∈ 𝒫 𝐽)
6 ispcmp 33856 . . . . . 6 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
7 pcmplfin.x . . . . . . 7 𝑋 = 𝐽
87iscref 33843 . . . . . 6 (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
96, 8bitri 275 . . . . 5 (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
109simprbi 496 . . . 4 (𝐽 ∈ Paracomp → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
11103ad2ant1 1134 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
12 simp3 1139 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑋 = 𝑈)
13 unieq 4918 . . . . . 6 (𝑢 = 𝑈 𝑢 = 𝑈)
1413eqeq2d 2748 . . . . 5 (𝑢 = 𝑈 → (𝑋 = 𝑢𝑋 = 𝑈))
15 breq2 5147 . . . . . 6 (𝑢 = 𝑈 → (𝑣Ref𝑢𝑣Ref𝑈))
1615rexbidv 3179 . . . . 5 (𝑢 = 𝑈 → (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢 ↔ ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈))
1714, 16imbi12d 344 . . . 4 (𝑢 = 𝑈 → ((𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) ↔ (𝑋 = 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
1817rspcv 3618 . . 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 4250 . 2 (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈 ↔ ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
2119, 20sylib 218 1 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  cfv 6561  Topctop 22899  Refcref 23510  LocFinclocfin 23512  CovHasRefccref 33841  Paracompcpcmp 33854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-cref 33842  df-pcmp 33855
This theorem is referenced by:  pcmplfinf  33860
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