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Theorem pcmplfin 30461
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 1171 . . . 4 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈𝐽)
2 ssexg 5029 . . . . . . 7 ((𝑈𝐽𝐽 ∈ Paracomp) → 𝑈 ∈ V)
32ancoms 452 . . . . . 6 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽) → 𝑈 ∈ V)
433adant3 1166 . . . . 5 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑈 ∈ V)
5 elpwg 4386 . . . . 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 30458 . . . . . 6 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
9 pcmplfin.x . . . . . . 7 𝑋 = 𝐽
109iscref 30445 . . . . . 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 1167 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
14 simp3 1172 . . 3 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → 𝑋 = 𝑈)
15 unieq 4666 . . . . . 6 (𝑢 = 𝑈 𝑢 = 𝑈)
1615eqeq2d 2835 . . . . 5 (𝑢 = 𝑈 → (𝑋 = 𝑢𝑋 = 𝑈))
17 breq2 4877 . . . . . 6 (𝑢 = 𝑈 → (𝑣Ref𝑢𝑣Ref𝑈))
1817rexbidv 3262 . . . . 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 3522 . . 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 4023 . . . . 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 3249 . 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 1111   = wceq 1656  wcel 2164  wral 3117  wrex 3118  Vcvv 3414  cin 3797  wss 3798  𝒫 cpw 4378   cuni 4658   class class class wbr 4873  cfv 6123  Topctop 21068  Refcref 21676  LocFinclocfin 21678  CovHasRefccref 30443  Paracompcpcmp 30456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-cref 30444  df-pcmp 30457
This theorem is referenced by:  pcmplfinf  30462
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