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Theorem pcmplfin 28761
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  = 
U. J
Assertion
Ref Expression
pcmplfin  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  E. v  e.  ~P  J ( v  e.  ( LocFin `  J
)  /\  v Ref U ) )
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 1031 . . . 4  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  U  C_  J )
2 ssexg 4542 . . . . . . 7  |-  ( ( U  C_  J  /\  J  e. Paracomp )  ->  U  e.  _V )
32ancoms 460 . . . . . 6  |-  ( ( J  e. Paracomp  /\  U  C_  J )  ->  U  e.  _V )
433adant3 1050 . . . . 5  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  U  e.  _V )
5 elpwg 3950 . . . . 5  |-  ( U  e.  _V  ->  ( U  e.  ~P J  <->  U 
C_  J ) )
64, 5syl 17 . . . 4  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  ( U  e.  ~P J  <->  U 
C_  J ) )
71, 6mpbird 240 . . 3  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  U  e.  ~P J )
8 ispcmp 28758 . . . . . 6  |-  ( J  e. Paracomp 
<->  J  e. CovHasRef ( LocFin `  J
) )
9 pcmplfin.x . . . . . . 7  |-  X  = 
U. J
109iscref 28745 . . . . . 6  |-  ( J  e. CovHasRef ( LocFin `  J )  <->  ( J  e.  Top  /\  A. u  e.  ~P  J
( X  =  U. u  ->  E. v  e.  ( ~P J  i^i  ( LocFin `
 J ) ) v Ref u ) ) )
118, 10bitri 257 . . . . 5  |-  ( J  e. Paracomp 
<->  ( J  e.  Top  /\ 
A. u  e.  ~P  J ( X  = 
U. u  ->  E. v  e.  ( ~P J  i^i  ( LocFin `  J )
) v Ref u
) ) )
1211simprbi 471 . . . 4  |-  ( J  e. Paracomp  ->  A. u  e.  ~P  J ( X  = 
U. u  ->  E. v  e.  ( ~P J  i^i  ( LocFin `  J )
) v Ref u
) )
13123ad2ant1 1051 . . 3  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  A. u  e.  ~P  J ( X  =  U. u  ->  E. v  e.  ( ~P J  i^i  ( LocFin `
 J ) ) v Ref u ) )
14 simp3 1032 . . 3  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  X  =  U. U )
15 unieq 4198 . . . . . 6  |-  ( u  =  U  ->  U. u  =  U. U )
1615eqeq2d 2481 . . . . 5  |-  ( u  =  U  ->  ( X  =  U. u  <->  X  =  U. U ) )
17 breq2 4399 . . . . . 6  |-  ( u  =  U  ->  (
v Ref u  <->  v Ref U ) )
1817rexbidv 2892 . . . . 5  |-  ( u  =  U  ->  ( E. v  e.  ( ~P J  i^i  ( LocFin `
 J ) ) v Ref u  <->  E. v  e.  ( ~P J  i^i  ( LocFin `  J )
) v Ref U
) )
1916, 18imbi12d 327 . . . 4  |-  ( u  =  U  ->  (
( X  =  U. u  ->  E. v  e.  ( ~P J  i^i  ( LocFin `
 J ) ) v Ref u )  <-> 
( X  =  U. U  ->  E. v  e.  ( ~P J  i^i  ( LocFin `
 J ) ) v Ref U ) ) )
2019rspcv 3132 . . 3  |-  ( U  e.  ~P J  -> 
( A. u  e. 
~P  J ( X  =  U. u  ->  E. v  e.  ( ~P J  i^i  ( LocFin `
 J ) ) v Ref u )  ->  ( X  = 
U. U  ->  E. v  e.  ( ~P J  i^i  ( LocFin `  J )
) v Ref U
) ) )
217, 13, 14, 20syl3c 62 . 2  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  E. v  e.  ( ~P J  i^i  ( LocFin `  J )
) v Ref U
)
22 elin 3608 . . . . 5  |-  ( v  e.  ( ~P J  i^i  ( LocFin `  J )
)  <->  ( v  e. 
~P J  /\  v  e.  ( LocFin `  J )
) )
2322anbi1i 709 . . . 4  |-  ( ( v  e.  ( ~P J  i^i  ( LocFin `  J ) )  /\  v Ref U )  <->  ( (
v  e.  ~P J  /\  v  e.  ( LocFin `
 J ) )  /\  v Ref U
) )
24 anass 661 . . . 4  |-  ( ( ( v  e.  ~P J  /\  v  e.  (
LocFin `  J ) )  /\  v Ref U
)  <->  ( v  e. 
~P J  /\  (
v  e.  ( LocFin `  J )  /\  v Ref U ) ) )
2523, 24bitri 257 . . 3  |-  ( ( v  e.  ( ~P J  i^i  ( LocFin `  J ) )  /\  v Ref U )  <->  ( v  e.  ~P J  /\  (
v  e.  ( LocFin `  J )  /\  v Ref U ) ) )
2625rexbii2 2879 . 2  |-  ( E. v  e.  ( ~P J  i^i  ( LocFin `  J ) ) v Ref U  <->  E. v  e.  ~P  J ( v  e.  ( LocFin `  J
)  /\  v Ref U ) )
2721, 26sylib 201 1  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  E. v  e.  ~P  J ( v  e.  ( LocFin `  J
)  /\  v Ref U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   class class class wbr 4395   ` cfv 5589   Topctop 19994   Refcref 20594   LocFinclocfin 20596  CovHasRefccref 28743  Paracompcpcmp 28756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-cref 28744  df-pcmp 28757
This theorem is referenced by:  pcmplfinf  28762
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