MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnei Structured version   Unicode version

Theorem isnei 19897
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 19896 . . 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 2472 . 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 3464 . . . . . . 7  |-  ( v  =  N  ->  (
g  C_  v  <->  g  C_  N ) )
54anbi2d 702 . . . . . 6  |-  ( v  =  N  ->  (
( S  C_  g  /\  g  C_  v )  <-> 
( S  C_  g  /\  g  C_  N ) ) )
65rexbidv 2918 . . . . 5  |-  ( v  =  N  ->  ( E. g  e.  J  ( S  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) ) )
76elrab 3207 . . . 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 19707 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  J )
9 elpw2g 4557 . . . . . 6  |-  ( X  e.  J  ->  ( N  e.  ~P X  <->  N 
C_  X ) )
108, 9syl 17 . . . . 5  |-  ( J  e.  Top  ->  ( N  e.  ~P X  <->  N 
C_  X ) )
1110anbi1d 703 . . . 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 463 . 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 367    = wceq 1405    e. wcel 1842   E.wrex 2755   {crab 2758    C_ wss 3414   ~Pcpw 3955   U.cuni 4191   ` cfv 5569   Topctop 19686   neicnei 19891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-top 19691  df-nei 19892
This theorem is referenced by:  neiint  19898  isneip  19899  neii1  19900  neii2  19902  neiss  19903  neips  19907  opnneissb  19908  opnssneib  19909  ssnei2  19910  innei  19919  neitr  19974  neitx  20400  neifg  30599  islptre  36993
  Copyright terms: Public domain W3C validator