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Theorem isneip 19474
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 4177 . . 3  |-  ( P  e.  X  ->  { P }  C_  X )
2 neifval.1 . . . 4  |-  X  = 
U. J
32isnei 19472 . . 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 4166 . . . . . 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 2978 . . . 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 1379    e. wcel 1767   E.wrex 2818    C_ wss 3481   {csn 4033   U.cuni 4251   ` cfv 5594   Topctop 19263   neicnei 19466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-top 19268  df-nei 19467
This theorem is referenced by:  neips  19482  neindisj  19486  neindisj2  19492  neiptopnei  19501  cnpnei  19633  fbflim2  20346  cnpflf2  20369  neibl  20872  neibastop2  30097  neibastop3  30098
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