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Theorem neival 12301
Description: Value of 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 12298 . . . 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 5416 . . 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 274 . 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 12164 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4076 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 14 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 295 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
9 pwexg 4099 . . . . 5  |-  ( X  e.  J  ->  ~P X  e.  _V )
10 rabexg 4066 . . . . 5  |-  ( ~P X  e.  _V  ->  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) }  e.  _V )
115, 9, 103syl 17 . . . 4  |-  ( J  e.  Top  ->  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) }  e.  _V )
1211adantr 274 . . 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 3115 . . . . . . 7  |-  ( x  =  S  ->  (
x  C_  g  <->  S  C_  g
) )
1413anbi1d 460 . . . . . 6  |-  ( x  =  S  ->  (
( x  C_  g  /\  g  C_  v )  <-> 
( S  C_  g  /\  g  C_  v ) ) )
1514rexbidv 2436 . . . . 5  |-  ( x  =  S  ->  ( E. g  e.  J  ( x  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) ) )
1615rabbidv 2670 . . . 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 2137 . . . 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 5490 . . 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 408 . 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 2170 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    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E.wrex 2415   {crab 2418   _Vcvv 2681    C_ wss 3066   ~Pcpw 3505   U.cuni 3731    |-> cmpt 3984   ` cfv 5118   Topctop 12153   neicnei 12296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-top 12154  df-nei 12297
This theorem is referenced by:  isnei  12302
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