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Theorem pcmplfin 28101
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 995 . . . 4  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  U  C_  J )
2 ssexg 4583 . . . . . . 7  |-  ( ( U  C_  J  /\  J  e. Paracomp )  ->  U  e.  _V )
32ancoms 451 . . . . . 6  |-  ( ( J  e. Paracomp  /\  U  C_  J )  ->  U  e.  _V )
433adant3 1014 . . . . 5  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  U  e.  _V )
5 elpwg 4007 . . . . 5  |-  ( U  e.  _V  ->  ( U  e.  ~P J  <->  U 
C_  J ) )
64, 5syl 16 . . . 4  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  ( U  e.  ~P J  <->  U 
C_  J ) )
71, 6mpbird 232 . . 3  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  U  e.  ~P J )
8 ispcmp 28098 . . . . . 6  |-  ( J  e. Paracomp 
<->  J  e. CovHasRef ( LocFin `  J
) )
9 pcmplfin.x . . . . . . 7  |-  X  = 
U. J
109iscref 28085 . . . . . 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 249 . . . . 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 462 . . . 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 1015 . . 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 996 . . 3  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  X  =  U. U )
15 unieq 4243 . . . . . 6  |-  ( u  =  U  ->  U. u  =  U. U )
1615eqeq2d 2468 . . . . 5  |-  ( u  =  U  ->  ( X  =  U. u  <->  X  =  U. U ) )
17 breq2 4443 . . . . . 6  |-  ( u  =  U  ->  (
v Ref u  <->  v Ref U ) )
1817rexbidv 2965 . . . . 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 318 . . . 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 3203 . . 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 61 . 2  |-  ( ( J  e. Paracomp  /\  U  C_  J  /\  X  =  U. U )  ->  E. v  e.  ( ~P J  i^i  ( LocFin `  J )
) v Ref U
)
22 elin 3673 . . . . 5  |-  ( v  e.  ( ~P J  i^i  ( LocFin `  J )
)  <->  ( v  e. 
~P J  /\  v  e.  ( LocFin `  J )
) )
2322anbi1i 693 . . . 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 647 . . . 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 249 . . 3  |-  ( ( v  e.  ( ~P J  i^i  ( LocFin `  J ) )  /\  v Ref U )  <->  ( v  e.  ~P J  /\  (
v  e.  ( LocFin `  J )  /\  v Ref U ) ) )
2625rexbii2 2954 . 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 196 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 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   class class class wbr 4439   ` cfv 5570   Topctop 19564   Refcref 20172   LocFinclocfin 20174  CovHasRefccref 28083  Paracompcpcmp 28096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-cref 28084  df-pcmp 28097
This theorem is referenced by:  pcmplfinf  28102
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