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Theorem locfindis 26387
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 26385 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  C  e.  PtFin )
2 unipw 4416 . . . . 5  |-  U. ~P X  =  X
32eqcomi 2442 . . . 4  |-  X  = 
U. ~P X
4 locfindis.1 . . . 4  |-  Y  = 
U. C
53, 4locfinbas 26383 . . 3  |-  ( C  e.  ( LocFin `  ~P X )  ->  X  =  Y )
61, 5jca 520 . 2  |-  ( C  e.  ( LocFin `  ~P X )  ->  ( C  e.  PtFin  /\  X  =  Y ) )
7 simpr 449 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  =  Y )
8 uniexg 4708 . . . . . . 7  |-  ( C  e.  PtFin  ->  U. C  e. 
_V )
94, 8syl5eqel 2522 . . . . . 6  |-  ( C  e.  PtFin  ->  Y  e.  _V )
109adantr 453 . . . . 5  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  Y  e.  _V )
117, 10eqeltrd 2512 . . . 4  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  X  e.  _V )
12 distop 17062 . . . 4  |-  ( X  e.  _V  ->  ~P X  e.  Top )
1311, 12syl 16 . . 3  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  ~P X  e.  Top )
14 snelpwi 4411 . . . . . 6  |-  ( x  e.  X  ->  { x }  e.  ~P X
)
1514adantl 454 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { x }  e.  ~P X
)
16 snidg 3841 . . . . . 6  |-  ( x  e.  X  ->  x  e.  { x } )
1716adantl 454 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  { x } )
18 simpll 732 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  C  e.  PtFin
)
197eleq2d 2505 . . . . . . 7  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  (
x  e.  X  <->  x  e.  Y ) )
2019biimpa 472 . . . . . 6  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  x  e.  Y )
214ptfinfin 26380 . . . . . 6  |-  ( ( C  e.  PtFin  /\  x  e.  Y )  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
2218, 20, 21syl2anc 644 . . . . 5  |-  ( ( ( C  e.  PtFin  /\  X  =  Y )  /\  x  e.  X
)  ->  { s  e.  C  |  x  e.  s }  e.  Fin )
23 eleq2 2499 . . . . . . 7  |-  ( y  =  { x }  ->  ( x  e.  y  <-> 
x  e.  { x } ) )
24 ineq2 3538 . . . . . . . . . . 11  |-  ( y  =  { x }  ->  ( s  i^i  y
)  =  ( s  i^i  { x }
) )
2524neeq1d 2616 . . . . . . . . . 10  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  ( s  i^i  { x } )  =/=  (/) ) )
26 disjsn 3870 . . . . . . . . . . 11  |-  ( ( s  i^i  { x } )  =  (/)  <->  -.  x  e.  s )
2726necon2abii 2661 . . . . . . . . . 10  |-  ( x  e.  s  <->  ( s  i^i  { x } )  =/=  (/) )
2825, 27syl6bbr 256 . . . . . . . . 9  |-  ( y  =  { x }  ->  ( ( s  i^i  y )  =/=  (/)  <->  x  e.  s ) )
2928rabbidv 2950 . . . . . . . 8  |-  ( y  =  { x }  ->  { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  =  { s  e.  C  |  x  e.  s } )
3029eleq1d 2504 . . . . . . 7  |-  ( y  =  { x }  ->  ( { s  e.  C  |  ( s  i^i  y )  =/=  (/) }  e.  Fin  <->  { s  e.  C  |  x  e.  s }  e.  Fin ) )
3123, 30anbi12d 693 . . . . . 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 3054 . . . . 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 1183 . . . 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 2791 . . 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 26378 . . 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 1139 . 2  |-  ( ( C  e.  PtFin  /\  X  =  Y )  ->  C  e.  ( LocFin `  ~P X ) )
376, 36impbii 182 1  |-  ( C  e.  ( LocFin `  ~P X )  <->  ( C  e.  PtFin  /\  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    i^i cin 3321   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017   ` cfv 5456   Fincfn 7111   Topctop 16960   PtFincptfin 26343   LocFinclocfin 26344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-er 6907  df-en 7112  df-fin 7115  df-top 16965  df-ptfin 26347  df-locfin 26348
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