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Theorem lfinpfin 20317
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 2402 . . . . . . . 8  |-  U. J  =  U. J
2 eqid 2402 . . . . . . . 8  |-  U. A  =  U. A
31, 2locfinbas 20315 . . . . . . 7  |-  ( A  e.  ( LocFin `  J
)  ->  U. J  = 
U. A )
43eleq2d 2472 . . . . . 6  |-  ( A  e.  ( LocFin `  J
)  ->  ( x  e.  U. J  <->  x  e.  U. A ) )
54biimpar 483 . . . . 5  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A )  ->  x  e.  U. J )
61locfinnei 20316 . . . . 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 468 . . . 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 3824 . . . . . . . . . 10  |-  ( ( x  e.  s  /\  x  e.  n )  ->  ( s  i^i  n
)  =/=  (/) )
98expcom 433 . . . . . . . . 9  |-  ( x  e.  n  ->  (
x  e.  s  -> 
( s  i^i  n
)  =/=  (/) ) )
109ad2antlr 725 . . . . . . . 8  |-  ( ( ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A
)  /\  x  e.  n )  /\  s  e.  A )  ->  (
x  e.  s  -> 
( s  i^i  n
)  =/=  (/) ) )
1110ss2rabdv 3520 . . . . . . 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 7775 . . . . . . . 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 433 . . . . . . 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 17 . . . . . 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 601 . . . . 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 2899 . . . 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 2818 . 2  |-  ( A  e.  ( LocFin `  J
)  ->  A. x  e.  U. A { s  e.  A  |  x  e.  s }  e.  Fin )
192isptfin 20309 . 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 367    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   {crab 2758    i^i cin 3413    C_ wss 3414   (/)c0 3738   U.cuni 4191   ` cfv 5569   Fincfn 7554   PtFincptfin 20296   LocFinclocfin 20297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-om 6684  df-er 7348  df-en 7555  df-fin 7558  df-top 19691  df-ptfin 20299  df-locfin 20300
This theorem is referenced by:  locfindis  20323
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