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Theorem locfindis 20600
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 20594 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  C  e.  PtFin )
2 unipw 4667 . . . . 5  |-  U. ~P X  =  X
32eqcomi 2471 . . . 4  |-  X  = 
U. ~P X
4 locfindis.1 . . . 4  |-  Y  = 
U. C
53, 4locfinbas 20592 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  X  =  Y )
61, 5jca 539 . 2  |-  ( C  e.  ( LocFin `  ~P X )  ->  ( C  e.  PtFin  /\  X  =  Y ) )
7 simpr 467 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  =  Y )
8 uniexg 6620 . . . . . . 7  |-  ( C  e.  PtFin  ->  U. C  e. 
_V )
94, 8syl5eqel 2544 . . . . . 6  |-  ( C  e.  PtFin  ->  Y  e.  _V )
109adantr 471 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  Y  e.  _V )
117, 10eqeltrd 2540 . . . 4  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  e.  _V )
12 distop 20066 . . . 4  |-  ( X  e.  _V  ->  ~P X  e.  Top )
1311, 12syl 17 . . 3  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  ~P X  e.  Top )
14 snelpwi 4662 . . . . . 6  |-  ( x  e.  X  ->  { x }  e.  ~P X
)
1514adantl 472 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { x }  e.  ~P X
)
16 snidg 4006 . . . . . 6  |-  ( x  e.  X  ->  x  e.  { x } )
1716adantl 472 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  { x } )
18 simpll 765 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  C  e.  PtFin
)
197eleq2d 2525 . . . . . . 7  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  (
x  e.  X  <->  x  e.  Y ) )
2019biimpa 491 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  Y )
214ptfinfin 20589 . . . . . 6  |-  ( ( C  e.  PtFin  /\  x  e.  Y )  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
2218, 20, 21syl2anc 671 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
23 eleq2 2529 . . . . . . 7  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
24 ineq2 3640 . . . . . . . . . . 11  |-  ( y  =  { x }  ->  ( s  i^i  y
)  =  ( s  i^i  { x }
) )
2524neeq1d 2695 . . . . . . . . . 10  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  ( s  i^i  { x } )  =/=  (/) ) )
26 disjsn 4044 . . . . . . . . . . 11  |-  ( ( s  i^i  { x } )  =  (/)  <->  -.  x  e.  s )
2726necon2abii 2686 . . . . . . . . . 10  |-  ( x  e.  s  <->  ( s  i^i  { x } )  =/=  (/) )
2825, 27syl6bbr 271 . . . . . . . . 9  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  x  e.  s ) )
2928rabbidv 3048 . . . . . . . 8  |-  ( y  =  { x }  ->  { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  =  { s  e.  C  |  x  e.  s } )
3029eleq1d 2524 . . . . . . 7  |-  ( y  =  { x }  ->  ( { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  e.  Fin  <->  { s  e.  C  |  x  e.  s }  e.  Fin ) )
3123, 30anbi12d 722 . . . . . 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 3162 . . . . 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 1274 . . . 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 2814 . . 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 20587 . . 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 1198 . 2  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  C  e.  ( LocFin `  ~P X ) )
376, 36impbii 192 1  |-  ( C  e.  ( LocFin `  ~P X )  <->  ( C  e.  PtFin  /\  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753   _Vcvv 3057    i^i cin 3415   (/)c0 3743   ~Pcpw 3963   {csn 3980   U.cuni 4212   ` cfv 5605   Fincfn 7600   Topctop 19972   PtFincptfin 20573   LocFinclocfin 20574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-om 6725  df-er 7394  df-en 7601  df-fin 7604  df-top 19976  df-ptfin 20576  df-locfin 20577
This theorem is referenced by: (None)
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