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Theorem locfindis 19897
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 19891 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  C  e.  PtFin )
2 unipw 4683 . . . . 5  |-  U. ~P X  =  X
32eqcomi 2454 . . . 4  |-  X  = 
U. ~P X
4 locfindis.1 . . . 4  |-  Y  = 
U. C
53, 4locfinbas 19889 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  X  =  Y )
61, 5jca 532 . 2  |-  ( C  e.  ( LocFin `  ~P X )  ->  ( C  e.  PtFin  /\  X  =  Y ) )
7 simpr 461 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  =  Y )
8 uniexg 6578 . . . . . . 7  |-  ( C  e.  PtFin  ->  U. C  e. 
_V )
94, 8syl5eqel 2533 . . . . . 6  |-  ( C  e.  PtFin  ->  Y  e.  _V )
109adantr 465 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  Y  e.  _V )
117, 10eqeltrd 2529 . . . 4  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  e.  _V )
12 distop 19363 . . . 4  |-  ( X  e.  _V  ->  ~P X  e.  Top )
1311, 12syl 16 . . 3  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  ~P X  e.  Top )
14 snelpwi 4678 . . . . . 6  |-  ( x  e.  X  ->  { x }  e.  ~P X
)
1514adantl 466 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { x }  e.  ~P X
)
16 snidg 4036 . . . . . 6  |-  ( x  e.  X  ->  x  e.  { x } )
1716adantl 466 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  { x } )
18 simpll 753 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  C  e.  PtFin
)
197eleq2d 2511 . . . . . . 7  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  (
x  e.  X  <->  x  e.  Y ) )
2019biimpa 484 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  Y )
214ptfinfin 19886 . . . . . 6  |-  ( ( C  e.  PtFin  /\  x  e.  Y )  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
2218, 20, 21syl2anc 661 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
23 eleq2 2514 . . . . . . 7  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
24 ineq2 3676 . . . . . . . . . . 11  |-  ( y  =  { x }  ->  ( s  i^i  y
)  =  ( s  i^i  { x }
) )
2524neeq1d 2718 . . . . . . . . . 10  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  ( s  i^i  { x } )  =/=  (/) ) )
26 disjsn 4071 . . . . . . . . . . 11  |-  ( ( s  i^i  { x } )  =  (/)  <->  -.  x  e.  s )
2726necon2abii 2707 . . . . . . . . . 10  |-  ( x  e.  s  <->  ( s  i^i  { x } )  =/=  (/) )
2825, 27syl6bbr 263 . . . . . . . . 9  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  x  e.  s ) )
2928rabbidv 3085 . . . . . . . 8  |-  ( y  =  { x }  ->  { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  =  { s  e.  C  |  x  e.  s } )
3029eleq1d 2510 . . . . . . 7  |-  ( y  =  { x }  ->  ( { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  e.  Fin  <->  { s  e.  C  |  x  e.  s }  e.  Fin ) )
3123, 30anbi12d 710 . . . . . 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 3194 . . . . 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 1225 . . . 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 2855 . . 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 19884 . . 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 1179 . 2  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  C  e.  ( LocFin `  ~P X ) )
376, 36impbii 188 1  |-  ( C  e.  ( LocFin `  ~P X )  <->  ( C  e.  PtFin  /\  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   E.wrex 2792   {crab 2795   _Vcvv 3093    i^i cin 3457   (/)c0 3767   ~Pcpw 3993   {csn 4010   U.cuni 4230   ` cfv 5574   Fincfn 7514   Topctop 19261   PtFincptfin 19870   LocFinclocfin 19871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-er 7309  df-en 7515  df-fin 7518  df-top 19266  df-ptfin 19873  df-locfin 19874
This theorem is referenced by: (None)
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