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Theorem neiss 19903
Description: Any neighborhood of a set  S is also a neighborhood of any subset  R  C_  S. Theorem of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)
Assertion
Ref Expression
neiss  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  e.  ( ( nei `  J
) `  R )
)

Proof of Theorem neiss
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . . . 4  |-  U. J  =  U. J
21neii1 19900 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  N  C_  U. J )
323adant3 1017 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  C_ 
U. J )
4 neii2 19902 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
543adant3 1017 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  E. g  e.  J  ( S  C_  g  /\  g  C_  N ) )
6 sstr2 3449 . . . . . 6  |-  ( R 
C_  S  ->  ( S  C_  g  ->  R  C_  g ) )
76anim1d 562 . . . . 5  |-  ( R 
C_  S  ->  (
( S  C_  g  /\  g  C_  N )  ->  ( R  C_  g  /\  g  C_  N
) ) )
87reximdv 2878 . . . 4  |-  ( R 
C_  S  ->  ( E. g  e.  J  ( S  C_  g  /\  g  C_  N )  ->  E. g  e.  J  ( R  C_  g  /\  g  C_  N ) ) )
983ad2ant3 1020 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  ( E. g  e.  J  ( S  C_  g  /\  g  C_  N )  ->  E. g  e.  J  ( R  C_  g  /\  g  C_  N ) ) )
105, 9mpd 15 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  E. g  e.  J  ( R  C_  g  /\  g  C_  N ) )
11 simp1 997 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  J  e.  Top )
12 simp3 999 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  R  C_  S )
131neiss2 19895 . . . . 5  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S ) )  ->  S  C_  U. J )
14133adant3 1017 . . . 4  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  S  C_ 
U. J )
1512, 14sstrd 3452 . . 3  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  R  C_ 
U. J )
161isnei 19897 . . 3  |-  ( ( J  e.  Top  /\  R  C_  U. J )  ->  ( N  e.  ( ( nei `  J
) `  R )  <->  ( N  C_  U. J  /\  E. g  e.  J  ( R  C_  g  /\  g  C_  N ) ) ) )
1711, 15, 16syl2anc 659 . 2  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  ( N  e.  ( ( nei `  J ) `  R )  <->  ( N  C_ 
U. J  /\  E. g  e.  J  ( R  C_  g  /\  g  C_  N ) ) ) )
183, 10, 17mpbir2and 923 1  |-  ( ( J  e.  Top  /\  N  e.  ( ( nei `  J ) `  S )  /\  R  C_  S )  ->  N  e.  ( ( nei `  J
) `  R )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    e. wcel 1842   E.wrex 2755    C_ wss 3414   U.cuni 4191   ` cfv 5569   Topctop 19686   neicnei 19891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-top 19691  df-nei 19892
This theorem is referenced by:  neips  19907  neissex  19921
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