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Theorem lfinpfin 26281
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 2412 . . . . . . . 8  |-  U. J  =  U. J
2 eqid 2412 . . . . . . . 8  |-  U. A  =  U. A
31, 2locfinbas 26279 . . . . . . 7  |-  ( A  e.  ( LocFin `  J
)  ->  U. J  = 
U. A )
43eleq2d 2479 . . . . . 6  |-  ( A  e.  ( LocFin `  J
)  ->  ( x  e.  U. J  <->  x  e.  U. A ) )
54biimpar 472 . . . . 5  |-  ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A )  ->  x  e.  U. J )
61locfinnei 26280 . . . . 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 457 . . . 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 3650 . . . . . . . . . 10  |-  ( ( x  e.  s  /\  x  e.  n )  ->  ( s  i^i  n
)  =/=  (/) )
98expcom 425 . . . . . . . . 9  |-  ( x  e.  n  ->  (
x  e.  s  -> 
( s  i^i  n
)  =/=  (/) ) )
109ad2antlr 708 . . . . . . . 8  |-  ( ( ( ( A  e.  ( LocFin `  J )  /\  x  e.  U. A
)  /\  x  e.  n )  /\  s  e.  A )  ->  (
x  e.  s  -> 
( s  i^i  n
)  =/=  (/) ) )
1110ss2rabdv 3392 . . . . . . 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 7296 . . . . . . . 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 425 . . . . . . 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 587 . . . . 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 2801 . . . 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 2757 . 2  |-  ( A  e.  ( LocFin `  J
)  ->  A. x  e.  U. A { s  e.  A  |  x  e.  s }  e.  Fin )
192isptfin 26273 . 2  |-  ( A  e.  ( LocFin `  J
)  ->  ( A  e.  PtFin 
<-> 
A. x  e.  U. A { s  e.  A  |  x  e.  s }  e.  Fin )
)
2018, 19mpbird 224 1  |-  ( A  e.  ( LocFin `  J
)  ->  A  e.  PtFin
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675   {crab 2678    i^i cin 3287    C_ wss 3288   (/)c0 3596   U.cuni 3983   ` cfv 5421   Fincfn 7076   PtFincptfin 26239   LocFinclocfin 26240
This theorem is referenced by:  locfindis  26283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-er 6872  df-en 7077  df-fin 7080  df-top 16926  df-ptfin 26243  df-locfin 26244
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