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Theorem locfindis 26305
Description: The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
locfindis.1  |-  Y  = 
U. C
Assertion
Ref Expression
locfindis  |-  ( C  e.  ( LocFin `  ~P X )  <->  ( C  e.  PtFin  /\  X  =  Y ) )

Proof of Theorem locfindis
Dummy variables  x  s  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfinpfin 26303 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  C  e.  PtFin )
2 unipw 4224 . . . . 5  |-  U. ~P X  =  X
32eqcomi 2287 . . . 4  |-  X  = 
U. ~P X
4 locfindis.1 . . . 4  |-  Y  = 
U. C
53, 4locfinbas 26301 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  X  =  Y )
61, 5jca 518 . 2  |-  ( C  e.  ( LocFin `  ~P X )  ->  ( C  e.  PtFin  /\  X  =  Y ) )
7 simpr 447 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  =  Y )
8 uniexg 4517 . . . . . . 7  |-  ( C  e.  PtFin  ->  U. C  e. 
_V )
94, 8syl5eqel 2367 . . . . . 6  |-  ( C  e.  PtFin  ->  Y  e.  _V )
109adantr 451 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  Y  e.  _V )
117, 10eqeltrd 2357 . . . 4  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  e.  _V )
12 distop 16733 . . . 4  |-  ( X  e.  _V  ->  ~P X  e.  Top )
1311, 12syl 15 . . 3  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  ~P X  e.  Top )
14 snelpwi 4220 . . . . . 6  |-  ( x  e.  X  ->  { x }  e.  ~P X
)
1514adantl 452 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { x }  e.  ~P X
)
16 snidg 3665 . . . . . 6  |-  ( x  e.  X  ->  x  e.  { x } )
1716adantl 452 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  { x } )
18 simpll 730 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  C  e.  PtFin
)
197eleq2d 2350 . . . . . . 7  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  (
x  e.  X  <->  x  e.  Y ) )
2019biimpa 470 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  Y )
214ptfinfin 26298 . . . . . 6  |-  ( ( C  e.  PtFin  /\  x  e.  Y )  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
2218, 20, 21syl2anc 642 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
23 eleq2 2344 . . . . . . 7  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
24 ineq2 3364 . . . . . . . . . . 11  |-  ( y  =  { x }  ->  ( s  i^i  y
)  =  ( s  i^i  { x }
) )
2524neeq1d 2459 . . . . . . . . . 10  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  ( s  i^i  { x } )  =/=  (/) ) )
26 disjsn 3693 . . . . . . . . . . 11  |-  ( ( s  i^i  { x } )  =  (/)  <->  -.  x  e.  s )
2726necon2abii 2501 . . . . . . . . . 10  |-  ( x  e.  s  <->  ( s  i^i  { x } )  =/=  (/) )
2825, 27syl6bbr 254 . . . . . . . . 9  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  x  e.  s ) )
2928rabbidv 2780 . . . . . . . 8  |-  ( y  =  { x }  ->  { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  =  { s  e.  C  |  x  e.  s } )
3029eleq1d 2349 . . . . . . 7  |-  ( y  =  { x }  ->  ( { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  e.  Fin  <->  { s  e.  C  |  x  e.  s }  e.  Fin ) )
3123, 30anbi12d 691 . . . . . 6  |-  ( y  =  { x }  ->  ( ( x  e.  y  /\  { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  e.  Fin ) 
<->  ( x  e.  {
x }  /\  {
s  e.  C  |  x  e.  s }  e.  Fin ) ) )
3231rspcev 2884 . . . . 5  |-  ( ( { x }  e.  ~P X  /\  (
x  e.  { x }  /\  { s  e.  C  |  x  e.  s }  e.  Fin ) )  ->  E. y  e.  ~P  X ( x  e.  y  /\  {
s  e.  C  | 
( s  i^i  y
)  =/=  (/) }  e.  Fin ) )
3315, 17, 22, 32syl12anc 1180 . . . 4  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  E. y  e.  ~P  X ( x  e.  y  /\  {
s  e.  C  | 
( s  i^i  y
)  =/=  (/) }  e.  Fin ) )
3433ralrimiva 2626 . . 3  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  A. x  e.  X  E. y  e.  ~P  X ( x  e.  y  /\  {
s  e.  C  | 
( s  i^i  y
)  =/=  (/) }  e.  Fin ) )
353, 4islocfin 26296 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  <->  ( ~P X  e.  Top  /\  X  =  Y  /\  A. x  e.  X  E. y  e.  ~P  X ( x  e.  y  /\  {
s  e.  C  | 
( s  i^i  y
)  =/=  (/) }  e.  Fin ) ) )
3613, 7, 34, 35syl3anbrc 1136 . 2  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  C  e.  ( LocFin `  ~P X ) )
376, 36impbii 180 1  |-  ( C  e.  ( LocFin `  ~P X )  <->  ( C  e.  PtFin  /\  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827   ` cfv 5255   Fincfn 6863   Topctop 16631   PtFincptfin 26261   LocFinclocfin 26262
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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