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Theorem pcmplfin 32909
Description: Given a paracompact topology 𝐽 and an open cover π‘ˆ, there exists an open refinement 𝑣 that is locally finite. (Contributed by Thierry Arnoux, 31-Jan-2020.)
Hypothesis
Ref Expression
pcmplfin.x 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
pcmplfin ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ))
Distinct variable groups:   𝑣,𝐽   𝑣,π‘ˆ
Allowed substitution hint:   𝑋(𝑣)
Proof of Theorem pcmplfin
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 ssexg 5323 . . . . . 6 ((π‘ˆ βŠ† 𝐽 ∧ 𝐽 ∈ Paracomp) β†’ π‘ˆ ∈ V)
21ancoms 459 . . . . 5 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽) β†’ π‘ˆ ∈ V)
323adant3 1132 . . . 4 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ π‘ˆ ∈ V)
4 simp2 1137 . . . 4 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ π‘ˆ βŠ† 𝐽)
53, 4elpwd 4608 . . 3 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ π‘ˆ ∈ 𝒫 𝐽)
6 ispcmp 32906 . . . . . 6 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½))
7 pcmplfin.x . . . . . . 7 𝑋 = βˆͺ 𝐽
87iscref 32893 . . . . . 6 (𝐽 ∈ CovHasRef(LocFinβ€˜π½) ↔ (𝐽 ∈ Top ∧ βˆ€π‘’ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑒 β†’ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Ref𝑒)))
96, 8bitri 274 . . . . 5 (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ βˆ€π‘’ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑒 β†’ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Ref𝑒)))
109simprbi 497 . . . 4 (𝐽 ∈ Paracomp β†’ βˆ€π‘’ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑒 β†’ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Ref𝑒))
11103ad2ant1 1133 . . 3 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆ€π‘’ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑒 β†’ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Ref𝑒))
12 simp3 1138 . . 3 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ 𝑋 = βˆͺ π‘ˆ)
13 unieq 4919 . . . . . 6 (𝑒 = π‘ˆ β†’ βˆͺ 𝑒 = βˆͺ π‘ˆ)
1413eqeq2d 2743 . . . . 5 (𝑒 = π‘ˆ β†’ (𝑋 = βˆͺ 𝑒 ↔ 𝑋 = βˆͺ π‘ˆ))
15 breq2 5152 . . . . . 6 (𝑒 = π‘ˆ β†’ (𝑣Ref𝑒 ↔ 𝑣Refπ‘ˆ))
1615rexbidv 3178 . . . . 5 (𝑒 = π‘ˆ β†’ (βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Ref𝑒 ↔ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Refπ‘ˆ))
1714, 16imbi12d 344 . . . 4 (𝑒 = π‘ˆ β†’ ((𝑋 = βˆͺ 𝑒 β†’ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Ref𝑒) ↔ (𝑋 = βˆͺ π‘ˆ β†’ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Refπ‘ˆ)))
1817rspcv 3608 . . 3 (π‘ˆ ∈ 𝒫 𝐽 β†’ (βˆ€π‘’ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑒 β†’ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Ref𝑒) β†’ (𝑋 = βˆͺ π‘ˆ β†’ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Refπ‘ˆ)))
195, 11, 12, 18syl3c 66 . 2 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Refπ‘ˆ)
20 rexin 4239 . 2 (βˆƒπ‘£ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑣Refπ‘ˆ ↔ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ))
2119, 20sylib 217 1 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908   class class class wbr 5148  β€˜cfv 6543  Topctop 22402  Refcref 23013  LocFinclocfin 23015  CovHasRefccref 32891  Paracompcpcmp 32904
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-cref 32892  df-pcmp 32905
This theorem is referenced by:  pcmplfinf  32910
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