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Theorem isnei 17092
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 17091 . . 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 2456 . 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 3315 . . . . . . 7  |-  ( v  =  N  ->  (
g  C_  v  <->  g  C_  N ) )
54anbi2d 685 . . . . . 6  |-  ( v  =  N  ->  (
( S  C_  g  /\  g  C_  v )  <-> 
( S  C_  g  /\  g  C_  N ) ) )
65rexbidv 2672 . . . . 5  |-  ( v  =  N  ->  ( E. g  e.  J  ( S  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
76elrab 3037 . . . 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 16904 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  J )
9 elpw2g 4306 . . . . . 6  |-  ( X  e.  J  ->  ( N  e.  ~P X  <->  N 
C_  X ) )
108, 9syl 16 . . . . 5  |-  ( J  e.  Top  ->  ( N  e.  ~P X  <->  N 
C_  X ) )
1110anbi1d 686 . . . 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 249 . . 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 452 . 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 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2652   {crab 2655    C_ wss 3265   ~Pcpw 3744   U.cuni 3959   ` cfv 5396   Topctop 16883   neicnei 17086
This theorem is referenced by:  neiint  17093  isneip  17094  neii1  17095  neii2  17097  neiss  17098  neips  17102  opnneissb  17103  opnssneib  17104  ssnei2  17105  innei  17114  neitr  17168  neitx  17562  neifg  26093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-top 16888  df-nei 17087
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