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Theorem lfinpfin 26303
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 2283 . . . . . . . 8  |-  U. J  =  U. J
2 eqid 2283 . . . . . . . 8  |-  U. A  =  U. A
31, 2locfinbas 26301 . . . . . . 7  |-  ( A  e.  ( LocFin `  J
)  ->  U. J  = 
U. A )
43eleq2d 2350 . . . . . 6  |-  ( A  e.  ( LocFin `  J
)  ->  ( x  e.  U. J  <->  x  e.  U. A ) )
54biimpar 471 . . . . 5  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A )  ->  x  e.  U. J )
61locfinnei 26302 . . . . 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 456 . . . 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 3509 . . . . . . . . . 10  |-  ( ( x  e.  s  /\  x  e.  n )  ->  ( s  i^i  n
)  =/=  (/) )
98expcom 424 . . . . . . . . 9  |-  ( x  e.  n  ->  (
x  e.  s  -> 
( s  i^i  n
)  =/=  (/) ) )
109ad2antlr 707 . . . . . . . 8  |-  ( ( ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A
)  /\  x  e.  n )  /\  s  e.  A )  ->  (
x  e.  s  -> 
( s  i^i  n
)  =/=  (/) ) )
1110ss2rabdv 3254 . . . . . . 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 7083 . . . . . . . 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 424 . . . . . . 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 15 . . . . . 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 586 . . . . 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 2670 . . . 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 14 . . 3  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A )  ->  { s  e.  A  |  x  e.  s }  e.  Fin )
1817ralrimiva 2626 . 2  |-  ( A  e.  ( LocFin `  J
)  ->  A. x  e.  U. A { s  e.  A  |  x  e.  s }  e.  Fin )
192isptfin 26295 . 2  |-  ( A  e.  ( LocFin `  J
)  ->  ( A  e.  PtFin 
<-> 
A. x  e.  U. A { s  e.  A  |  x  e.  s }  e.  Fin )
)
2018, 19mpbird 223 1  |-  ( A  e.  ( LocFin `  J
)  ->  A  e.  PtFin
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   ` cfv 5255   Fincfn 6863   PtFincptfin 26261   LocFinclocfin 26262
This theorem is referenced by:  locfindis  26305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-fin 6867  df-top 16636  df-ptfin 26265  df-locfin 26266
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