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Theorem isneip 19584
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 4159 . . 3  |-  ( P  e.  X  ->  { P }  C_  X )
2 neifval.1 . . . 4  |-  X  = 
U. J
32isnei 19582 . . 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 4148 . . . . . 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 2954 . . . 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 1383    e. wcel 1804   E.wrex 2794    C_ wss 3461   {csn 4014   U.cuni 4234   ` cfv 5578   Topctop 19372   neicnei 19576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-top 19377  df-nei 19577
This theorem is referenced by:  neips  19592  neindisj  19596  neindisj2  19602  neiptopnei  19611  cnpnei  19743  fbflim2  20456  cnpflf2  20479  neibl  20982  neibastop2  30155  neibastop3  30156
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