Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lfinpfin Structured version   Unicode version

Theorem lfinpfin 28710
Description: A locally finite cover is point-finite. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Assertion
Ref Expression
lfinpfin  |-  ( A  e.  ( LocFin `  J
)  ->  A  e.  PtFin
)

Proof of Theorem lfinpfin
Dummy variables  n  s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2451 . . . . . . . 8  |-  U. J  =  U. J
2 eqid 2451 . . . . . . . 8  |-  U. A  =  U. A
31, 2locfinbas 28708 . . . . . . 7  |-  ( A  e.  ( LocFin `  J
)  ->  U. J  = 
U. A )
43eleq2d 2520 . . . . . 6  |-  ( A  e.  ( LocFin `  J
)  ->  ( x  e.  U. J  <->  x  e.  U. A ) )
54biimpar 485 . . . . 5  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A )  ->  x  e.  U. J )
61locfinnei 28709 . . . . 5  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. J )  ->  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
75, 6syldan 470 . . . 4  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A )  ->  E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin ) )
8 inelcm 3828 . . . . . . . . . 10  |-  ( ( x  e.  s  /\  x  e.  n )  ->  ( s  i^i  n
)  =/=  (/) )
98expcom 435 . . . . . . . . 9  |-  ( x  e.  n  ->  (
x  e.  s  -> 
( s  i^i  n
)  =/=  (/) ) )
109ad2antlr 726 . . . . . . . 8  |-  ( ( ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A
)  /\  x  e.  n )  /\  s  e.  A )  ->  (
x  e.  s  -> 
( s  i^i  n
)  =/=  (/) ) )
1110ss2rabdv 3528 . . . . . . 7  |-  ( ( ( A  e.  (
LocFin `  J )  /\  x  e.  U. A )  /\  x  e.  n
)  ->  { s  e.  A  |  x  e.  s }  C_  { s  e.  A  |  ( s  i^i  n )  =/=  (/) } )
12 ssfi 7631 . . . . . . . 8  |-  ( ( { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin  /\  { s  e.  A  |  x  e.  s }  C_  { s  e.  A  |  ( s  i^i  n )  =/=  (/) } )  ->  { s  e.  A  |  x  e.  s }  e.  Fin )
1312expcom 435 . . . . . . 7  |-  ( { s  e.  A  |  x  e.  s }  C_ 
{ s  e.  A  |  ( s  i^i  n )  =/=  (/) }  ->  ( { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin  ->  { s  e.  A  |  x  e.  s }  e.  Fin ) )
1411, 13syl 16 . . . . . 6  |-  ( ( ( A  e.  (
LocFin `  J )  /\  x  e.  U. A )  /\  x  e.  n
)  ->  ( {
s  e.  A  | 
( s  i^i  n
)  =/=  (/) }  e.  Fin  ->  { s  e.  A  |  x  e.  s }  e.  Fin ) )
1514expimpd 603 . . . . 5  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A )  -> 
( ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )  ->  { s  e.  A  |  x  e.  s }  e.  Fin ) )
1615rexlimdvw 2937 . . . 4  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A )  -> 
( E. n  e.  J  ( x  e.  n  /\  { s  e.  A  |  ( s  i^i  n )  =/=  (/) }  e.  Fin )  ->  { s  e.  A  |  x  e.  s }  e.  Fin ) )
177, 16mpd 15 . . 3  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A )  ->  { s  e.  A  |  x  e.  s }  e.  Fin )
1817ralrimiva 2820 . 2  |-  ( A  e.  ( LocFin `  J
)  ->  A. x  e.  U. A { s  e.  A  |  x  e.  s }  e.  Fin )
192isptfin 28702 . 2  |-  ( A  e.  ( LocFin `  J
)  ->  ( A  e.  PtFin 
<-> 
A. x  e.  U. A { s  e.  A  |  x  e.  s }  e.  Fin )
)
2018, 19mpbird 232 1  |-  ( A  e.  ( LocFin `  J
)  ->  A  e.  PtFin
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758    =/= wne 2642   A.wral 2793   E.wrex 2794   {crab 2797    i^i cin 3422    C_ wss 3423   (/)c0 3732   U.cuni 4186   ` cfv 5513   Fincfn 7407   PtFincptfin 28668   LocFinclocfin 28669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-om 6574  df-er 7198  df-en 7408  df-fin 7411  df-top 18616  df-ptfin 28672  df-locfin 28673
This theorem is referenced by:  locfindis  28712
  Copyright terms: Public domain W3C validator