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Theorem pcmplfin 29751
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 1060 . . . 4 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈𝐽)
2 ssexg 4774 . . . . . . 7 ((𝑈𝐽𝐽 ∈ Paracomp) → 𝑈 ∈ V)
32ancoms 469 . . . . . 6 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽) → 𝑈 ∈ V)
433adant3 1079 . . . . 5 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈 ∈ V)
5 elpwg 4144 . . . . 5 (𝑈 ∈ V → (𝑈 ∈ 𝒫 𝐽𝑈𝐽))
64, 5syl 17 . . . 4 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → (𝑈 ∈ 𝒫 𝐽𝑈𝐽))
71, 6mpbird 247 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈 ∈ 𝒫 𝐽)
8 ispcmp 29748 . . . . . 6 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
9 pcmplfin.x . . . . . . 7 𝑋 = 𝐽
109iscref 29735 . . . . . 6 (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
118, 10bitri 264 . . . . 5 (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
1211simprbi 480 . . . 4 (𝐽 ∈ Paracomp → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
13123ad2ant1 1080 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
14 simp3 1061 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑋 = 𝑈)
15 unieq 4417 . . . . . 6 (𝑢 = 𝑈 𝑢 = 𝑈)
1615eqeq2d 2631 . . . . 5 (𝑢 = 𝑈 → (𝑋 = 𝑢𝑋 = 𝑈))
17 breq2 4627 . . . . . 6 (𝑢 = 𝑈 → (𝑣Ref𝑢𝑣Ref𝑈))
1817rexbidv 3047 . . . . 5 (𝑢 = 𝑈 → (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢 ↔ ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈))
1916, 18imbi12d 334 . . . 4 (𝑢 = 𝑈 → ((𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) ↔ (𝑋 = 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
2019rspcv 3295 . . 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 3780 . . . . 5 (𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ↔ (𝑣 ∈ 𝒫 𝐽𝑣 ∈ (LocFin‘𝐽)))
2322anbi1i 730 . . . 4 ((𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈) ↔ ((𝑣 ∈ 𝒫 𝐽𝑣 ∈ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈))
24 anass 680 . . . 4 (((𝑣 ∈ 𝒫 𝐽𝑣 ∈ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈) ↔ (𝑣 ∈ 𝒫 𝐽 ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)))
2523, 24bitri 264 . . 3 ((𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈) ↔ (𝑣 ∈ 𝒫 𝐽 ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)))
2625rexbii2 3034 . 2 (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈 ↔ ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
2721, 26sylib 208 1 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wrex 2909  Vcvv 3190  cin 3559  wss 3560  𝒫 cpw 4136   cuni 4409   class class class wbr 4623  cfv 5857  Topctop 20638  Refcref 21245  LocFinclocfin 21247  CovHasRefccref 29733  Paracompcpcmp 29746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-cref 29734  df-pcmp 29747
This theorem is referenced by:  pcmplfinf  29752
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