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Theorem dissnlocfin 20008
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 19475 . 2  |-  ( X  e.  V  ->  ~P X  e.  Top )
2 eqidd 2444 . 2  |-  ( X  e.  V  ->  X  =  X )
3 snelpwi 4682 . . . . 5  |-  ( z  e.  X  ->  { z }  e.  ~P X
)
43adantl 466 . . . 4  |-  ( ( X  e.  V  /\  z  e.  X )  ->  { z }  e.  ~P X )
5 ssnid 4043 . . . . 5  |-  z  e. 
{ z }
65a1i 11 . . . 4  |-  ( ( X  e.  V  /\  z  e.  X )  ->  z  e.  { z } )
7 nfv 1694 . . . . . 6  |-  F/ u
( X  e.  V  /\  z  e.  X
)
8 nfrab1 3024 . . . . . 6  |-  F/_ u { u  e.  C  |  ( u  i^i 
{ z } )  =/=  (/) }
9 nfcv 2605 . . . . . 6  |-  F/_ u { { z } }
10 dissnref.c . . . . . . . . . 10  |-  C  =  { u  |  E. x  e.  X  u  =  { x } }
1110abeq2i 2570 . . . . . . . . 9  |-  ( u  e.  C  <->  E. x  e.  X  u  =  { x } )
1211anbi1i 695 . . . . . . . 8  |-  ( ( u  e.  C  /\  ( u  i^i  { z } )  =/=  (/) )  <->  ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) ) )
13 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  u  =  { x } )
14 simplr 755 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  u  =  { x } )
1514ineq1d 3684 . . . . . . . . . . . . . . . . 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 4076 . . . . . . . . . . . . . . . . . 18  |-  ( x  =/=  z  ->  ( { x }  i^i  { z } )  =  (/) )
1716adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  ( {
x }  i^i  {
z } )  =  (/) )
1815, 17eqtrd 2484 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  ( u  i^i  { z } )  =/=  (/) )
2019neneqd 2645 . . . . . . . . . . . . . . . 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 576 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  -.  x  =/=  z
)
22 nne 2644 . . . . . . . . . . . . . . 15  |-  ( -.  x  =/=  z  <->  x  =  z )
2321, 22sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  x  =  z )
2423sneqd 4026 . . . . . . . . . . . . 13  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  { x }  =  { z } )
2513, 24eqtrd 2484 . . . . . . . . . . . 12  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  u  =  { z } )
2625r19.29an 2984 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  E. x  e.  X  u  =  { x }
)  ->  u  =  { z } )
2726an32s 804 . . . . . . . . . 10  |-  ( ( ( ( X  e.  V  /\  z  e.  X )  /\  E. x  e.  X  u  =  { x } )  /\  ( u  i^i 
{ z } )  =/=  (/) )  ->  u  =  { z } )
2827anasss 647 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) ) )  ->  u  =  {
z } )
29 sneq 4024 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  { x }  =  { z } )
3029eqeq2d 2457 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
u  =  { x } 
<->  u  =  { z } ) )
3130rspcev 3196 . . . . . . . . . . 11  |-  ( ( z  e.  X  /\  u  =  { z } )  ->  E. x  e.  X  u  =  { x } )
3231adantll 713 . . . . . . . . . 10  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  E. x  e.  X  u  =  { x } )
33 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  u  =  {
z } )
3433ineq1d 3684 . . . . . . . . . . . 12  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( u  i^i 
{ z } )  =  ( { z }  i^i  { z } ) )
35 inidm 3692 . . . . . . . . . . . 12  |-  ( { z }  i^i  {
z } )  =  { z }
3634, 35syl6eq 2500 . . . . . . . . . . 11  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( u  i^i 
{ z } )  =  { z } )
37 vex 3098 . . . . . . . . . . . . 13  |-  z  e. 
_V
3837snnz 4133 . . . . . . . . . . . 12  |-  { z }  =/=  (/)
3938a1i 11 . . . . . . . . . . 11  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  { z }  =/=  (/) )
4036, 39eqnetrd 2736 . . . . . . . . . 10  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( u  i^i 
{ z } )  =/=  (/) )
4132, 40jca 532 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) ) )
4228, 41impbida 832 . . . . . . . 8  |-  ( ( X  e.  V  /\  z  e.  X )  ->  ( ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) )  <->  u  =  { z } ) )
4312, 42syl5bb 257 . . . . . . 7  |-  ( ( X  e.  V  /\  z  e.  X )  ->  ( ( u  e.  C  /\  ( u  i^i  { z } )  =/=  (/) )  <->  u  =  { z } ) )
44 rabid 3020 . . . . . . 7  |-  ( u  e.  { u  e.  C  |  ( u  i^i  { z } )  =/=  (/) }  <->  ( u  e.  C  /\  (
u  i^i  { z } )  =/=  (/) ) )
45 elsn 4028 . . . . . . 7  |-  ( u  e.  { { z } }  <->  u  =  { z } )
4643, 44, 453bitr4g 288 . . . . . 6  |-  ( ( X  e.  V  /\  z  e.  X )  ->  ( u  e.  {
u  e.  C  | 
( u  i^i  {
z } )  =/=  (/) }  <->  u  e.  { {
z } } ) )
477, 8, 9, 46eqrd 3507 . . . . 5  |-  ( ( X  e.  V  /\  z  e.  X )  ->  { u  e.  C  |  ( u  i^i 
{ z } )  =/=  (/) }  =  { { z } }
)
48 snfi 7598 . . . . 5  |-  { {
z } }  e.  Fin
4947, 48syl6eqel 2539 . . . 4  |-  ( ( X  e.  V  /\  z  e.  X )  ->  { u  e.  C  |  ( u  i^i 
{ z } )  =/=  (/) }  e.  Fin )
50 eleq2 2516 . . . . . 6  |-  ( y  =  { z }  ->  ( z  e.  y  <->  z  e.  {
z } ) )
51 ineq2 3679 . . . . . . . . 9  |-  ( y  =  { z }  ->  ( u  i^i  y )  =  ( u  i^i  { z } ) )
5251neeq1d 2720 . . . . . . . 8  |-  ( y  =  { z }  ->  ( ( u  i^i  y )  =/=  (/) 
<->  ( u  i^i  {
z } )  =/=  (/) ) )
5352rabbidv 3087 . . . . . . 7  |-  ( y  =  { z }  ->  { u  e.  C  |  ( u  i^i  y )  =/=  (/) }  =  { u  e.  C  |  (
u  i^i  { z } )  =/=  (/) } )
5453eleq1d 2512 . . . . . 6  |-  ( y  =  { z }  ->  ( { u  e.  C  |  (
u  i^i  y )  =/=  (/) }  e.  Fin  <->  {
u  e.  C  | 
( u  i^i  {
z } )  =/=  (/) }  e.  Fin )
)
5550, 54anbi12d 710 . . . . 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 3196 . . . 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 1227 . . 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 2857 . 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 4687 . . . 4  |-  U. ~P X  =  X
6059eqcomi 2456 . . 3  |-  X  = 
U. ~P X
6110unisngl 20006 . . 3  |-  X  = 
U. C
6260, 61islocfin 19996 . 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 1181 1  |-  ( X  e.  V  ->  C  e.  ( LocFin `  ~P X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   {cab 2428    =/= wne 2638   A.wral 2793   E.wrex 2794   {crab 2797    i^i cin 3460   (/)c0 3770   ~Pcpw 3997   {csn 4014   U.cuni 4234   ` cfv 5578   Fincfn 7518   Topctop 19372   LocFinclocfin 19983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-1o 7132  df-en 7519  df-fin 7522  df-top 19377  df-locfin 19986
This theorem is referenced by:  dispcmp  27840
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