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Theorem isneip 18708
Description: The predicate " N is a neighborhood of point  P." (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
neifval.1  |-  X  = 
U. J
Assertion
Ref Expression
isneip  |-  ( ( J  e.  Top  /\  P  e.  X )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_  X  /\  E. g  e.  J  ( P  e.  g  /\  g  C_  N ) ) ) )
Distinct variable groups:    g, J    g, N    P, g    g, X

Proof of Theorem isneip
StepHypRef Expression
1 snssi 4016 . . 3  |-  ( P  e.  X  ->  { P }  C_  X )
2 neifval.1 . . . 4  |-  X  = 
U. J
32isnei 18706 . . 3  |-  ( ( J  e.  Top  /\  { P }  C_  X
)  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_  X  /\  E. g  e.  J  ( { P }  C_  g  /\  g  C_  N ) ) ) )
41, 3sylan2 474 . 2  |-  ( ( J  e.  Top  /\  P  e.  X )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_  X  /\  E. g  e.  J  ( { P }  C_  g  /\  g  C_  N ) ) ) )
5 snssg 4006 . . . . . 6  |-  ( P  e.  X  ->  ( P  e.  g  <->  { P }  C_  g ) )
65anbi1d 704 . . . . 5  |-  ( P  e.  X  ->  (
( P  e.  g  /\  g  C_  N
)  <->  ( { P }  C_  g  /\  g  C_  N ) ) )
76rexbidv 2735 . . . 4  |-  ( P  e.  X  ->  ( E. g  e.  J  ( P  e.  g  /\  g  C_  N )  <->  E. g  e.  J  ( { P }  C_  g  /\  g  C_  N
) ) )
87anbi2d 703 . . 3  |-  ( P  e.  X  ->  (
( N  C_  X  /\  E. g  e.  J  ( P  e.  g  /\  g  C_  N ) )  <->  ( N  C_  X  /\  E. g  e.  J  ( { P }  C_  g  /\  g  C_  N ) ) ) )
98adantl 466 . 2  |-  ( ( J  e.  Top  /\  P  e.  X )  ->  ( ( N  C_  X  /\  E. g  e.  J  ( P  e.  g  /\  g  C_  N ) )  <->  ( N  C_  X  /\  E. g  e.  J  ( { P }  C_  g  /\  g  C_  N ) ) ) )
104, 9bitr4d 256 1  |-  ( ( J  e.  Top  /\  P  e.  X )  ->  ( N  e.  ( ( nei `  J
) `  { P } )  <->  ( N  C_  X  /\  E. g  e.  J  ( P  e.  g  /\  g  C_  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2715    C_ wss 3327   {csn 3876   U.cuni 4090   ` cfv 5417   Topctop 18497   neicnei 18700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-top 18502  df-nei 18701
This theorem is referenced by:  neips  18716  neindisj  18720  neindisj2  18726  neiptopnei  18735  cnpnei  18867  fbflim2  19549  cnpflf2  19572  neibl  20075  neibastop2  28580  neibastop3  28581
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