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Theorem isnei 12156
Description: The predicate "the class  N is a neighborhood of  S". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
neifval.1  |-  X  = 
U. J
Assertion
Ref Expression
isnei  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
Distinct variable groups:    g, J    g, N    S, g    g, X

Proof of Theorem isnei
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . 4  |-  X  = 
U. J
21neival 12155 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( nei `  J
) `  S )  =  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) } )
32eleq2d 2184 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  N  e.  { v  e. 
~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) } ) )
4 sseq2 3087 . . . . . . 7  |-  ( v  =  N  ->  (
g  C_  v  <->  g  C_  N ) )
54anbi2d 457 . . . . . 6  |-  ( v  =  N  ->  (
( S  C_  g  /\  g  C_  v )  <-> 
( S  C_  g  /\  g  C_  N ) ) )
65rexbidv 2412 . . . . 5  |-  ( v  =  N  ->  ( E. g  e.  J  ( S  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
76elrab 2809 . . . 4  |-  ( N  e.  { v  e. 
~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) }  <->  ( N  e.  ~P X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
81topopn 12018 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  J )
9 elpw2g 4041 . . . . . 6  |-  ( X  e.  J  ->  ( N  e.  ~P X  <->  N 
C_  X ) )
108, 9syl 14 . . . . 5  |-  ( J  e.  Top  ->  ( N  e.  ~P X  <->  N 
C_  X ) )
1110anbi1d 458 . . . 4  |-  ( J  e.  Top  ->  (
( N  e.  ~P X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N
) )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
127, 11syl5bb 191 . . 3  |-  ( J  e.  Top  ->  ( N  e.  { v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) }  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
1312adantr 272 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  {
v  e.  ~P X  |  E. g  e.  J  ( S  C_  g  /\  g  C_  v ) }  <-> 
( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
143, 13bitrd 187 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( N  e.  ( ( nei `  J
) `  S )  <->  ( N  C_  X  /\  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2391   {crab 2394    C_ wss 3037   ~Pcpw 3476   U.cuni 3702   ` cfv 5081   Topctop 12007   neicnei 12150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-top 12008  df-nei 12151
This theorem is referenced by:  neiint  12157  isneip  12158  neii1  12159  neii2  12161  neiss  12162  neipsm  12166  opnneissb  12167  opnssneib  12168  ssnei2  12169  innei  12175  neitx  12279
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