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Theorem neival 16839
Description: The set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
neifval.1  |-  X  = 
U. J
Assertion
Ref Expression
neival  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( nei `  J
) `  S )  =  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) } )
Distinct variable groups:    v, g, J    S, g, v    g, X, v

Proof of Theorem neival
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . 5  |-  X  = 
U. J
21neifval 16836 . . . 4  |-  ( J  e.  Top  ->  ( nei `  J )  =  ( x  e.  ~P X  |->  { v  e. 
~P X  |  E. g  e.  J  (
x  C_  g  /\  g  C_  v ) } ) )
32fveq1d 5527 . . 3  |-  ( J  e.  Top  ->  (
( nei `  J
) `  S )  =  ( ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } ) `  S ) )
43adantr 451 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( nei `  J
) `  S )  =  ( ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } ) `  S ) )
51topopn 16652 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4174 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 15 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 471 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
9 pwexg 4194 . . . . 5  |-  ( X  e.  J  ->  ~P X  e.  _V )
10 rabexg 4164 . . . . 5  |-  ( ~P X  e.  _V  ->  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) }  e.  _V )
115, 9, 103syl 18 . . . 4  |-  ( J  e.  Top  ->  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) }  e.  _V )
1211adantr 451 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) }  e.  _V )
13 sseq1 3199 . . . . . . 7  |-  ( x  =  S  ->  (
x  C_  g  <->  S  C_  g
) )
1413anbi1d 685 . . . . . 6  |-  ( x  =  S  ->  (
( x  C_  g  /\  g  C_  v )  <-> 
( S  C_  g  /\  g  C_  v ) ) )
1514rexbidv 2564 . . . . 5  |-  ( x  =  S  ->  ( E. g  e.  J  ( x  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) ) )
1615rabbidv 2780 . . . 4  |-  ( x  =  S  ->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) }  =  { v  e. 
~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) } )
17 eqid 2283 . . . 4  |-  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )  =  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )
1816, 17fvmptg 5600 . . 3  |-  ( ( S  e.  ~P X  /\  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) }  e.  _V )  ->  ( ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } ) `
 S )  =  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) } )
198, 12, 18syl2anc 642 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( x  e. 
~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } ) `  S )  =  { v  e. 
~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) } )
204, 19eqtrd 2315 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( nei `  J
) `  S )  =  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U.cuni 3827    e. cmpt 4077   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem is referenced by:  isnei  16840  sallnei  25529
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-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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