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Theorem locfindis 20482
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 20476 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  C  e.  PtFin )
2 unipw 4663 . . . . 5  |-  U. ~P X  =  X
32eqcomi 2433 . . . 4  |-  X  = 
U. ~P X
4 locfindis.1 . . . 4  |-  Y  = 
U. C
53, 4locfinbas 20474 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  X  =  Y )
61, 5jca 534 . 2  |-  ( C  e.  ( LocFin `  ~P X )  ->  ( C  e.  PtFin  /\  X  =  Y ) )
7 simpr 462 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  =  Y )
8 uniexg 6593 . . . . . . 7  |-  ( C  e.  PtFin  ->  U. C  e. 
_V )
94, 8syl5eqel 2512 . . . . . 6  |-  ( C  e.  PtFin  ->  Y  e.  _V )
109adantr 466 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  Y  e.  _V )
117, 10eqeltrd 2508 . . . 4  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  e.  _V )
12 distop 19948 . . . 4  |-  ( X  e.  _V  ->  ~P X  e.  Top )
1311, 12syl 17 . . 3  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  ~P X  e.  Top )
14 snelpwi 4658 . . . . . 6  |-  ( x  e.  X  ->  { x }  e.  ~P X
)
1514adantl 467 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { x }  e.  ~P X
)
16 snidg 4019 . . . . . 6  |-  ( x  e.  X  ->  x  e.  { x } )
1716adantl 467 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  { x } )
18 simpll 758 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  C  e.  PtFin
)
197eleq2d 2490 . . . . . . 7  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  (
x  e.  X  <->  x  e.  Y ) )
2019biimpa 486 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  Y )
214ptfinfin 20471 . . . . . 6  |-  ( ( C  e.  PtFin  /\  x  e.  Y )  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
2218, 20, 21syl2anc 665 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
23 eleq2 2493 . . . . . . 7  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
24 ineq2 3655 . . . . . . . . . . 11  |-  ( y  =  { x }  ->  ( s  i^i  y
)  =  ( s  i^i  { x }
) )
2524neeq1d 2699 . . . . . . . . . 10  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  ( s  i^i  { x } )  =/=  (/) ) )
26 disjsn 4054 . . . . . . . . . . 11  |-  ( ( s  i^i  { x } )  =  (/)  <->  -.  x  e.  s )
2726necon2abii 2688 . . . . . . . . . 10  |-  ( x  e.  s  <->  ( s  i^i  { x } )  =/=  (/) )
2825, 27syl6bbr 266 . . . . . . . . 9  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  x  e.  s ) )
2928rabbidv 3070 . . . . . . . 8  |-  ( y  =  { x }  ->  { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  =  { s  e.  C  |  x  e.  s } )
3029eleq1d 2489 . . . . . . 7  |-  ( y  =  { x }  ->  ( { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  e.  Fin  <->  { s  e.  C  |  x  e.  s }  e.  Fin ) )
3123, 30anbi12d 715 . . . . . 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 3179 . . . . 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 1262 . . . 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 2837 . . 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 20469 . . 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 1189 . 2  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  C  e.  ( LocFin `  ~P X ) )
376, 36impbii 190 1  |-  ( C  e.  ( LocFin `  ~P X )  <->  ( C  e.  PtFin  /\  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   E.wrex 2774   {crab 2777   _Vcvv 3078    i^i cin 3432   (/)c0 3758   ~Pcpw 3976   {csn 3993   U.cuni 4213   ` cfv 5592   Fincfn 7568   Topctop 19854   PtFincptfin 20455   LocFinclocfin 20456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-er 7362  df-en 7569  df-fin 7572  df-top 19858  df-ptfin 20458  df-locfin 20459
This theorem is referenced by: (None)
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