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Theorem isnei 18840
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 18839 . . 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 2524 . 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 3487 . . . . . . 7  |-  ( v  =  N  ->  (
g  C_  v  <->  g  C_  N ) )
54anbi2d 703 . . . . . 6  |-  ( v  =  N  ->  (
( S  C_  g  /\  g  C_  v )  <-> 
( S  C_  g  /\  g  C_  N ) ) )
65rexbidv 2868 . . . . 5  |-  ( v  =  N  ->  ( E. g  e.  J  ( S  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
76elrab 3224 . . . 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 18652 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  J )
9 elpw2g 4564 . . . . . 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 704 . . . 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 257 . . 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 465 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800   {crab 2803    C_ wss 3437   ~Pcpw 3969   U.cuni 4200   ` cfv 5527   Topctop 18631   neicnei 18834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-top 18636  df-nei 18835
This theorem is referenced by:  neiint  18841  isneip  18842  neii1  18843  neii2  18845  neiss  18846  neips  18850  opnneissb  18851  opnssneib  18852  ssnei2  18853  innei  18862  neitr  18917  neitx  19313  neifg  28741
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