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Theorem dissnlocfin 20134
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 19601 . 2  |-  ( X  e.  V  ->  ~P X  e.  Top )
2 eqidd 2393 . 2  |-  ( X  e.  V  ->  X  =  X )
3 snelpwi 4620 . . . . 5  |-  ( z  e.  X  ->  { z }  e.  ~P X
)
43adantl 464 . . . 4  |-  ( ( X  e.  V  /\  z  e.  X )  ->  { z }  e.  ~P X )
5 ssnid 3986 . . . . 5  |-  z  e. 
{ z }
65a1i 11 . . . 4  |-  ( ( X  e.  V  /\  z  e.  X )  ->  z  e.  { z } )
7 nfv 1722 . . . . . 6  |-  F/ u
( X  e.  V  /\  z  e.  X
)
8 nfrab1 2976 . . . . . 6  |-  F/_ u { u  e.  C  |  ( u  i^i 
{ z } )  =/=  (/) }
9 nfcv 2554 . . . . . 6  |-  F/_ u { { z } }
10 dissnref.c . . . . . . . . . 10  |-  C  =  { u  |  E. x  e.  X  u  =  { x } }
1110abeq2i 2519 . . . . . . . . 9  |-  ( u  e.  C  <->  E. x  e.  X  u  =  { x } )
1211anbi1i 693 . . . . . . . 8  |-  ( ( u  e.  C  /\  ( u  i^i  { z } )  =/=  (/) )  <->  ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) ) )
13 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  u  =  { x } )
14 simplr 753 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  u  =  { x } )
1514ineq1d 3626 . . . . . . . . . . . . . . . . 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 4018 . . . . . . . . . . . . . . . . . 18  |-  ( x  =/=  z  ->  ( { x }  i^i  { z } )  =  (/) )
1716adantl 464 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  ( {
x }  i^i  {
z } )  =  (/) )
1815, 17eqtrd 2433 . . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  ( u  i^i  {
z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  { x } )  /\  x  =/=  z
)  ->  ( u  i^i  { z } )  =/=  (/) )
2019neneqd 2594 . . . . . . . . . . . . . . . 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 574 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  -.  x  =/=  z
)
22 nne 2593 . . . . . . . . . . . . . . 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 3969 . . . . . . . . . . . . 13  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  { x }  =  { z } )
2513, 24eqtrd 2433 . . . . . . . . . . . 12  |-  ( ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  x  e.  X )  /\  u  =  {
x } )  ->  u  =  { z } )
2625r19.29an 2936 . . . . . . . . . . 11  |-  ( ( ( ( X  e.  V  /\  z  e.  X )  /\  (
u  i^i  { z } )  =/=  (/) )  /\  E. x  e.  X  u  =  { x }
)  ->  u  =  { z } )
2726an32s 802 . . . . . . . . . 10  |-  ( ( ( ( X  e.  V  /\  z  e.  X )  /\  E. x  e.  X  u  =  { x } )  /\  ( u  i^i 
{ z } )  =/=  (/) )  ->  u  =  { z } )
2827anasss 645 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) ) )  ->  u  =  {
z } )
29 sneq 3967 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  { x }  =  { z } )
3029eqeq2d 2406 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
u  =  { x } 
<->  u  =  { z } ) )
3130rspcev 3148 . . . . . . . . . . 11  |-  ( ( z  e.  X  /\  u  =  { z } )  ->  E. x  e.  X  u  =  { x } )
3231adantll 711 . . . . . . . . . 10  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  E. x  e.  X  u  =  { x } )
33 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  u  =  {
z } )
3433ineq1d 3626 . . . . . . . . . . . 12  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( u  i^i 
{ z } )  =  ( { z }  i^i  { z } ) )
35 inidm 3634 . . . . . . . . . . . 12  |-  ( { z }  i^i  {
z } )  =  { z }
3634, 35syl6eq 2449 . . . . . . . . . . 11  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( u  i^i 
{ z } )  =  { z } )
37 vex 3050 . . . . . . . . . . . . 13  |-  z  e. 
_V
3837snnz 4075 . . . . . . . . . . . 12  |-  { z }  =/=  (/)
3938a1i 11 . . . . . . . . . . 11  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  { z }  =/=  (/) )
4036, 39eqnetrd 2685 . . . . . . . . . 10  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( u  i^i 
{ z } )  =/=  (/) )
4132, 40jca 530 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  z  e.  X
)  /\  u  =  { z } )  ->  ( E. x  e.  X  u  =  { x }  /\  ( u  i^i  { z } )  =/=  (/) ) )
4228, 41impbida 830 . . . . . . . 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 2972 . . . . . . 7  |-  ( u  e.  { u  e.  C  |  ( u  i^i  { z } )  =/=  (/) }  <->  ( u  e.  C  /\  (
u  i^i  { z } )  =/=  (/) ) )
45 elsn 3971 . . . . . . 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 3448 . . . . 5  |-  ( ( X  e.  V  /\  z  e.  X )  ->  { u  e.  C  |  ( u  i^i 
{ z } )  =/=  (/) }  =  { { z } }
)
48 snfi 7533 . . . . 5  |-  { {
z } }  e.  Fin
4947, 48syl6eqel 2488 . . . 4  |-  ( ( X  e.  V  /\  z  e.  X )  ->  { u  e.  C  |  ( u  i^i 
{ z } )  =/=  (/) }  e.  Fin )
50 eleq2 2465 . . . . . 6  |-  ( y  =  { z }  ->  ( z  e.  y  <->  z  e.  {
z } ) )
51 ineq2 3621 . . . . . . . . 9  |-  ( y  =  { z }  ->  ( u  i^i  y )  =  ( u  i^i  { z } ) )
5251neeq1d 2669 . . . . . . . 8  |-  ( y  =  { z }  ->  ( ( u  i^i  y )  =/=  (/) 
<->  ( u  i^i  {
z } )  =/=  (/) ) )
5352rabbidv 3039 . . . . . . 7  |-  ( y  =  { z }  ->  { u  e.  C  |  ( u  i^i  y )  =/=  (/) }  =  { u  e.  C  |  (
u  i^i  { z } )  =/=  (/) } )
5453eleq1d 2461 . . . . . 6  |-  ( y  =  { z }  ->  ( { u  e.  C  |  (
u  i^i  y )  =/=  (/) }  e.  Fin  <->  {
u  e.  C  | 
( u  i^i  {
z } )  =/=  (/) }  e.  Fin )
)
5550, 54anbi12d 708 . . . . 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 3148 . . . 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 1224 . . 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 2806 . 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 4625 . . . 4  |-  U. ~P X  =  X
6059eqcomi 2405 . . 3  |-  X  = 
U. ~P X
6110unisngl 20132 . . 3  |-  X  = 
U. C
6260, 61islocfin 20122 . 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 1178 1  |-  ( X  e.  V  ->  C  e.  ( LocFin `  ~P X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   {cab 2377    =/= wne 2587   A.wral 2742   E.wrex 2743   {crab 2746    i^i cin 3401   (/)c0 3724   ~Pcpw 3940   {csn 3957   U.cuni 4176   ` cfv 5509   Fincfn 7453   Topctop 19498   LocFinclocfin 20109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-om 6618  df-1o 7066  df-en 7454  df-fin 7457  df-top 19503  df-locfin 20112
This theorem is referenced by:  dispcmp  28047
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