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Theorem isneip 16898
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 3796 . . 3  |-  ( P  e.  X  ->  { P }  C_  X )
2 neifval.1 . . . 4  |-  X  = 
U. J
32isnei 16896 . . 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 460 . 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 3788 . . . . . 6  |-  ( P  e.  X  ->  ( P  e.  g  <->  { P }  C_  g ) )
65anbi1d 685 . . . . 5  |-  ( P  e.  X  ->  (
( P  e.  g  /\  g  C_  N
)  <->  ( { P }  C_  g  /\  g  C_  N ) ) )
76rexbidv 2598 . . . 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 684 . . 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 452 . 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 247 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 176    /\ wa 358    = wceq 1633    e. wcel 1701   E.wrex 2578    C_ wss 3186   {csn 3674   U.cuni 3864   ` cfv 5292   Topctop 16687   neicnei 16890
This theorem is referenced by:  neips  16906  neindisj  16910  neindisj2  16916  cnpnei  17049  fbflim2  17724  cnpflf2  17747  neibl  18099  neibastop2  25459  neibastop3  25460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-top 16692  df-nei 16891
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