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Theorem isnei 16840
Description: The predicate " 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 16839 . . 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 2350 . 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 3200 . . . . . . 7  |-  ( v  =  N  ->  (
g  C_  v  <->  g  C_  N ) )
54anbi2d 684 . . . . . 6  |-  ( v  =  N  ->  (
( S  C_  g  /\  g  C_  v )  <-> 
( S  C_  g  /\  g  C_  N ) ) )
65rexbidv 2564 . . . . 5  |-  ( v  =  N  ->  ( E. g  e.  J  ( S  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
76elrab 2923 . . . 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 16652 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  J )
9 elpw2g 4174 . . . . . 6  |-  ( X  e.  J  ->  ( N  e.  ~P X  <->  N 
C_  X ) )
108, 9syl 15 . . . . 5  |-  ( J  e.  Top  ->  ( N  e.  ~P X  <->  N 
C_  X ) )
1110anbi1d 685 . . . 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 248 . . 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 451 . 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 244 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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   ` cfv 5255   Topctop 16631   neicnei 16834
This theorem is referenced by:  neiint  16841  isneip  16842  neii1  16843  neii2  16845  neiss  16846  neips  16850  opnneissb  16851  opnssneib  16852  ssnei2  16853  innei  16862  osneisi  25531  neifg  26320
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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