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Theorem dissnlocfin 20542
Description: The set of singletons is locally finite in the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.)
Hypothesis
Ref Expression
dissnref.c  |-  C  =  { u  |  E. x  e.  X  u  =  { x } }
Assertion
Ref Expression
dissnlocfin  |-  ( X  e.  V  ->  C  e.  ( LocFin `  ~P X ) )
Distinct variable groups:    u, C, x    u, V, x    u, X, x

Proof of Theorem dissnlocfin
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 20009 . 2  |-  ( X  e.  V  ->  ~P X  e.  Top )
2 eqidd 2423 . 2  |-  ( X  e.  V  ->  X  =  X )
3 snelpwi 4666 . . . . 5  |-  ( z  e.  X  ->  { z }  e.  ~P X
)
43adantl 467 . . . 4  |-  ( ( X  e.  V  /\  z  e.  X )  ->  { z }  e.  ~P X )
5 ssnid 4027 . . . . 5  |-  z  e. 
{ z }
65a1i 11 . . . 4  |-  ( ( X  e.  V  /\  z  e.  X )  ->  z  e.  { z } )
7 nfv 1755 . . . . . 6  |-  F/ u
( X  e.  V  /\  z  e.  X
)
8 nfrab1 3006 . . . . . 6  |-  F/_ u { u  e.  C  |  ( u  i^i 
{ z } )  =/=  (/) }
9 nfcv 2580 . . . . . 6  |-  F/_ u { { z } }
10 dissnref.c . . . . . . . . . 10  |-  C  =  { u  |  E. x  e.  X  u  =  { x } }
1110abeq2i 2544 . . . . . . . . 9  |-  ( u  e.  C  <->  E. x  e.  X  u  =  { x } )
1211anbi1i 699 . . . . . . . 8  |-  ( ( u  e.  C  /\  ( u  i^i  { z } )  =/=  (/) )  <->  ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) ) )
13 simpr 462 . . . . . . . . . . . . 13  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  u  =  { x } )
14 simplr 760 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  u  =  { x } )
1514ineq1d 3663 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  ( u  i^i  { z } )  =  ( { x }  i^i  { z } ) )
16 disjsn2 4061 . . . . . . . . . . . . . . . . . 18  |-  ( x  =/=  z  ->  ( { x }  i^i  { z } )  =  (/) )
1716adantl 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  ( {
x }  i^i  {
z } )  =  (/) )
1815, 17eqtrd 2463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  ( u  i^i  { z } )  =  (/) )
19 simp-4r 775 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  ( u  i^i  { z } )  =/=  (/) )
2019neneqd 2621 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  -.  (
u  i^i  { z } )  =  (/) )
2118, 20pm2.65da 578 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  -.  x  =/=  z
)
22 nne 2620 . . . . . . . . . . . . . . 15  |-  ( -.  x  =/=  z  <->  x  =  z )
2321, 22sylib 199 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  x  =  z )
2423sneqd 4010 . . . . . . . . . . . . 13  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  { x }  =  { z } )
2513, 24eqtrd 2463 . . . . . . . . . . . 12  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  u  =  { z } )
2625r19.29an 2966 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  E. x  e.  X  u  =  { x }
)  ->  u  =  { z } )
2726an32s 811 . . . . . . . . . 10  |-  ( ( ( ( X  e.  V  /\  z  e.  X )  /\  E. x  e.  X  u  =  { x } )  /\  ( u  i^i 
{ z } )  =/=  (/) )  ->  u  =  { z } )
2827anasss 651 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) ) )  ->  u  =  {
z } )
29 sneq 4008 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  { x }  =  { z } )
3029eqeq2d 2436 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
u  =  { x } 
<->  u  =  { z } ) )
3130rspcev 3182 . . . . . . . . . . 11  |-  ( ( z  e.  X  /\  u  =  { z } )  ->  E. x  e.  X  u  =  { x } )
3231adantll 718 . . . . . . . . . 10  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  E. x  e.  X  u  =  { x } )
33 simpr 462 . . . . . . . . . . . . 13  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  u  =  {
z } )
3433ineq1d 3663 . . . . . . . . . . . 12  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( u  i^i 
{ z } )  =  ( { z }  i^i  { z } ) )
35 inidm 3671 . . . . . . . . . . . 12  |-  ( { z }  i^i  {
z } )  =  { z }
3634, 35syl6eq 2479 . . . . . . . . . . 11  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( u  i^i 
{ z } )  =  { z } )
37 vex 3083 . . . . . . . . . . . . 13  |-  z  e. 
_V
3837snnz 4118 . . . . . . . . . . . 12  |-  { z }  =/=  (/)
3938a1i 11 . . . . . . . . . . 11  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  { z }  =/=  (/) )
4036, 39eqnetrd 2713 . . . . . . . . . 10  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( u  i^i 
{ z } )  =/=  (/) )
4132, 40jca 534 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) ) )
4228, 41impbida 840 . . . . . . . 8  |-  ( ( X  e.  V  /\  z  e.  X )  ->  ( ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) )  <->  u  =  { z } ) )
4312, 42syl5bb 260 . . . . . . 7  |-  ( ( X  e.  V  /\  z  e.  X )  ->  ( ( u  e.  C  /\  ( u  i^i  { z } )  =/=  (/) )  <->  u  =  { z } ) )
44 rabid 3002 . . . . . . 7  |-  ( u  e.  { u  e.  C  |  ( u  i^i  { z } )  =/=  (/) }  <->  ( u  e.  C  /\  (
u  i^i  { z } )  =/=  (/) ) )
45 elsn 4012 . . . . . . 7  |-  ( u  e.  { { z } }  <->  u  =  { z } )
4643, 44, 453bitr4g 291 . . . . . 6  |-  ( ( X  e.  V  /\  z  e.  X )  ->  ( u  e.  {
u  e.  C  | 
( u  i^i  {
z } )  =/=  (/) }  <->  u  e.  { {
z } } ) )
477, 8, 9, 46eqrd 3482 . . . . 5  |-  ( ( X  e.  V  /\  z  e.  X )  ->  { u  e.  C  |  ( u  i^i 
{ z } )  =/=  (/) }  =  { { z } }
)
48 snfi 7660 . . . . 5  |-  { {
z } }  e.  Fin
4947, 48syl6eqel 2515 . . . 4  |-  ( ( X  e.  V  /\  z  e.  X )  ->  { u  e.  C  |  ( u  i^i 
{ z } )  =/=  (/) }  e.  Fin )
50 eleq2 2496 . . . . . 6  |-  ( y  =  { z }  ->  ( z  e.  y  <->  z  e.  {
z } ) )
51 ineq2 3658 . . . . . . . . 9  |-  ( y  =  { z }  ->  ( u  i^i  y )  =  ( u  i^i  { z } ) )
5251neeq1d 2697 . . . . . . . 8  |-  ( y  =  { z }  ->  ( ( u  i^i  y )  =/=  (/) 
<->  ( u  i^i  {
z } )  =/=  (/) ) )
5352rabbidv 3071 . . . . . . 7  |-  ( y  =  { z }  ->  { u  e.  C  |  ( u  i^i  y )  =/=  (/) }  =  { u  e.  C  |  (
u  i^i  { z } )  =/=  (/) } )
5453eleq1d 2491 . . . . . 6  |-  ( y  =  { z }  ->  ( { u  e.  C  |  (
u  i^i  y )  =/=  (/) }  e.  Fin  <->  {
u  e.  C  | 
( u  i^i  {
z } )  =/=  (/) }  e.  Fin )
)
5550, 54anbi12d 715 . . . . 5  |-  ( y  =  { z }  ->  ( ( z  e.  y  /\  {
u  e.  C  | 
( u  i^i  y
)  =/=  (/) }  e.  Fin )  <->  ( z  e. 
{ z }  /\  { u  e.  C  | 
( u  i^i  {
z } )  =/=  (/) }  e.  Fin )
) )
5655rspcev 3182 . . . 4  |-  ( ( { z }  e.  ~P X  /\  (
z  e.  { z }  /\  { u  e.  C  |  (
u  i^i  { z } )  =/=  (/) }  e.  Fin ) )  ->  E. y  e.  ~P  X ( z  e.  y  /\  {
u  e.  C  | 
( u  i^i  y
)  =/=  (/) }  e.  Fin ) )
574, 6, 49, 56syl12anc 1262 . . 3  |-  ( ( X  e.  V  /\  z  e.  X )  ->  E. y  e.  ~P  X ( z  e.  y  /\  { u  e.  C  |  (
u  i^i  y )  =/=  (/) }  e.  Fin ) )
5857ralrimiva 2836 . 2  |-  ( X  e.  V  ->  A. z  e.  X  E. y  e.  ~P  X ( z  e.  y  /\  {
u  e.  C  | 
( u  i^i  y
)  =/=  (/) }  e.  Fin ) )
59 unipw 4671 . . . 4  |-  U. ~P X  =  X
6059eqcomi 2435 . . 3  |-  X  = 
U. ~P X
6110unisngl 20540 . . 3  |-  X  = 
U. C
6260, 61islocfin 20530 . 2  |-  ( C  e.  ( LocFin `  ~P X )  <->  ( ~P X  e.  Top  /\  X  =  X  /\  A. z  e.  X  E. y  e.  ~P  X ( z  e.  y  /\  {
u  e.  C  | 
( u  i^i  y
)  =/=  (/) }  e.  Fin ) ) )
631, 2, 58, 62syl3anbrc 1189 1  |-  ( X  e.  V  ->  C  e.  ( LocFin `  ~P X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2407    =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775    i^i cin 3435   (/)c0 3761   ~Pcpw 3981   {csn 3998   U.cuni 4219   ` cfv 5601   Fincfn 7580   Topctop 19915   LocFinclocfin 20517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-1o 7193  df-en 7581  df-fin 7584  df-top 19919  df-locfin 20520
This theorem is referenced by:  dispcmp  28694
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