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Theorem nsn 25530
Description: The neighborhoods of the singletons are neighborhoods. (Contributed by FL, 2-Aug-2009.)
Hypothesis
Ref Expression
nsn.1  |-  X  = 
U. J
Assertion
Ref Expression
nsn  |-  ( J  e.  Top  ->  U_ x  e.  X  ( ( nei `  J ) `  { x } ) 
C_  U. ran  ( nei `  J ) )
Distinct variable group:    x, J
Allowed substitution hint:    X( x)

Proof of Theorem nsn
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eliun 3909 . . . . 5  |-  ( y  e.  U_ x  e.  X  ( ( nei `  J ) `  {
x } )  <->  E. x  e.  X  y  e.  ( ( nei `  J
) `  { x } ) )
2 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
32snelpw 4221 . . . . . . . . 9  |-  ( x  e.  X  <->  { x }  e.  ~P X
)
4 nsn.1 . . . . . . . . . . 11  |-  X  = 
U. J
54neif 16837 . . . . . . . . . 10  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
6 fndm 5343 . . . . . . . . . 10  |-  ( ( nei `  J )  Fn  ~P X  ->  dom  ( nei `  J
)  =  ~P X
)
7 eleq2 2344 . . . . . . . . . . . 12  |-  ( ~P X  =  dom  ( nei `  J )  -> 
( { x }  e.  ~P X  <->  { x }  e.  dom  ( nei `  J ) ) )
87biimpd 198 . . . . . . . . . . 11  |-  ( ~P X  =  dom  ( nei `  J )  -> 
( { x }  e.  ~P X  ->  { x }  e.  dom  ( nei `  J ) ) )
98eqcoms 2286 . . . . . . . . . 10  |-  ( dom  ( nei `  J
)  =  ~P X  ->  ( { x }  e.  ~P X  ->  { x }  e.  dom  ( nei `  J ) ) )
105, 6, 93syl 18 . . . . . . . . 9  |-  ( J  e.  Top  ->  ( { x }  e.  ~P X  ->  { x }  e.  dom  ( nei `  J ) ) )
113, 10syl5bi 208 . . . . . . . 8  |-  ( J  e.  Top  ->  (
x  e.  X  ->  { x }  e.  dom  ( nei `  J
) ) )
12 fveq2 5525 . . . . . . . . . . 11  |-  ( z  =  { x }  ->  ( ( nei `  J
) `  z )  =  ( ( nei `  J ) `  {
x } ) )
1312eleq2d 2350 . . . . . . . . . 10  |-  ( z  =  { x }  ->  ( y  e.  ( ( nei `  J
) `  z )  <->  y  e.  ( ( nei `  J ) `  {
x } ) ) )
1413rspcev 2884 . . . . . . . . 9  |-  ( ( { x }  e.  dom  ( nei `  J
)  /\  y  e.  ( ( nei `  J
) `  { x } ) )  ->  E. z  e.  dom  ( nei `  J ) y  e.  ( ( nei `  J ) `
 z ) )
1514ex 423 . . . . . . . 8  |-  ( { x }  e.  dom  ( nei `  J )  ->  ( y  e.  ( ( nei `  J
) `  { x } )  ->  E. z  e.  dom  ( nei `  J
) y  e.  ( ( nei `  J
) `  z )
) )
1611, 15syl6 29 . . . . . . 7  |-  ( J  e.  Top  ->  (
x  e.  X  -> 
( y  e.  ( ( nei `  J
) `  { x } )  ->  E. z  e.  dom  ( nei `  J
) y  e.  ( ( nei `  J
) `  z )
) ) )
1716com3l 75 . . . . . 6  |-  ( x  e.  X  ->  (
y  e.  ( ( nei `  J ) `
 { x }
)  ->  ( J  e.  Top  ->  E. z  e.  dom  ( nei `  J
) y  e.  ( ( nei `  J
) `  z )
) ) )
1817rexlimiv 2661 . . . . 5  |-  ( E. x  e.  X  y  e.  ( ( nei `  J ) `  {
x } )  -> 
( J  e.  Top  ->  E. z  e.  dom  ( nei `  J ) y  e.  ( ( nei `  J ) `
 z ) ) )
191, 18sylbi 187 . . . 4  |-  ( y  e.  U_ x  e.  X  ( ( nei `  J ) `  {
x } )  -> 
( J  e.  Top  ->  E. z  e.  dom  ( nei `  J ) y  e.  ( ( nei `  J ) `
 z ) ) )
2019com12 27 . . 3  |-  ( J  e.  Top  ->  (
y  e.  U_ x  e.  X  ( ( nei `  J ) `  { x } )  ->  E. z  e.  dom  ( nei `  J ) y  e.  ( ( nei `  J ) `
 z ) ) )
21 eqid 2283 . . . . 5  |-  U. J  =  U. J
2221neif 16837 . . . 4  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P U. J )
23 fnfun 5341 . . . 4  |-  ( ( nei `  J )  Fn  ~P U. J  ->  Fun  ( nei `  J
) )
24 elunirn 5777 . . . 4  |-  ( Fun  ( nei `  J
)  ->  ( y  e.  U. ran  ( nei `  J )  <->  E. z  e.  dom  ( nei `  J
) y  e.  ( ( nei `  J
) `  z )
) )
2522, 23, 243syl 18 . . 3  |-  ( J  e.  Top  ->  (
y  e.  U. ran  ( nei `  J )  <->  E. z  e.  dom  ( nei `  J ) y  e.  ( ( nei `  J ) `
 z ) ) )
2620, 25sylibrd 225 . 2  |-  ( J  e.  Top  ->  (
y  e.  U_ x  e.  X  ( ( nei `  J ) `  { x } )  ->  y  e.  U. ran  ( nei `  J
) ) )
2726ssrdv 3185 1  |-  ( J  e.  Top  ->  U_ x  e.  X  ( ( nei `  J ) `  { x } ) 
C_  U. ran  ( nei `  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   U_ciun 3905   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-nei 16835
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