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Theorem isneip 17169
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 3942 . . 3  |-  ( P  e.  X  ->  { P }  C_  X )
2 neifval.1 . . . 4  |-  X  = 
U. J
32isnei 17167 . . 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 461 . 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 3932 . . . . . 6  |-  ( P  e.  X  ->  ( P  e.  g  <->  { P }  C_  g ) )
65anbi1d 686 . . . . 5  |-  ( P  e.  X  ->  (
( P  e.  g  /\  g  C_  N
)  <->  ( { P }  C_  g  /\  g  C_  N ) ) )
76rexbidv 2726 . . . 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 685 . . 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 453 . 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 248 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    C_ wss 3320   {csn 3814   U.cuni 4015   ` cfv 5454   Topctop 16958   neicnei 17161
This theorem is referenced by:  neips  17177  neindisj  17181  neindisj2  17187  neiptopnei  17196  cnpnei  17328  fbflim2  18009  cnpflf2  18032  neibl  18531  neibastop2  26390  neibastop3  26391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-top 16963  df-nei 17162
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